34:I[651451,["/_next/static/chunks/2um_u-bxbfm5w.js?dpl=nogit-1792116633","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792116633","/_next/static/chunks/0z347qy8pf1j3.js?dpl=nogit-1792116633","/_next/static/chunks/02eggfg7vwd9q.js?dpl=nogit-1792116633","/_next/static/chunks/1gm-hv-3-fp7_.js?dpl=nogit-1792116633","/_next/static/chunks/2xa0kylife8l8.js?dpl=nogit-1792116633","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792116633","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792116633","/_next/static/chunks/0kyh6u8hfv06h.js?dpl=nogit-1792116633","/_next/static/chunks/1zirpvj720iz0.js?dpl=nogit-1792116633","/_next/static/chunks/0mt9yp27gb-6w.js?dpl=nogit-1792116633","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792116633","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792116633","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1792116633","/_next/static/chunks/1jo7uanvci4-r.js?dpl=nogit-1792116633","/_next/static/chunks/0papbjd-vx5q5.js?dpl=nogit-1792116633","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792116633","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792116633","/_next/static/chunks/23f2tb1apskc8.js?dpl=nogit-1792116633","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792116633","/_next/static/chunks/3x5t_ry1ziv7r.js?dpl=nogit-1792116633","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792116633","/_next/static/chunks/0iapw94rt37n4.js?dpl=nogit-1792116633","/_next/static/chunks/2rvalcou1gw7v.js?dpl=nogit-1792116633","/_next/static/chunks/1c-rbkxmrcq7-.js?dpl=nogit-1792116633","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792116633","/_next/static/chunks/04wg1hnm8knmc.js?dpl=nogit-1792116633","/_next/static/chunks/3_six85qevnm3.js?dpl=nogit-1792116633","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792116633","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792116633","/_next/static/chunks/1vovlfobo0rf7.js?dpl=nogit-1792116633","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792116633","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1792116633","/_next/static/chunks/1jxfzuqocprar.js?dpl=nogit-1792116633","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792116633","/_next/static/chunks/3bplvoggfmxkd.js?dpl=nogit-1792116633","/_next/static/chunks/1drwxxqzvnao9.js?dpl=nogit-1792116633","/_next/static/chunks/0ltoscgl22itf.js?dpl=nogit-1792116633","/_next/static/chunks/442ru-jd_6vlx.js?dpl=nogit-1792116633","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1792116633","/_next/static/chunks/1zo1304bjzjl_.js?dpl=nogit-1792116633","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1792116633","/_next/static/chunks/0beaxlbowx15e.js?dpl=nogit-1792116633","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1792116633","/_next/static/chunks/2mbz851h6l6ul.js?dpl=nogit-1792116633","/_next/static/chunks/2yss1g0cimuwe.js?dpl=nogit-1792116633","/_next/static/chunks/0ioq6d8hla7h0.js?dpl=nogit-1792116633","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792116633","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792116633","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792116633","/_next/static/chunks/421rwkxv431i9.js?dpl=nogit-1792116633","/_next/static/chunks/2l63yczi1z2qz.js?dpl=nogit-1792116633","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1792116633","/_next/static/chunks/33y-ad8441jub.js?dpl=nogit-1792116633","/_next/static/chunks/1iyn56mfdoj9b.js?dpl=nogit-1792116633","/_next/static/chunks/1d4d88jx43_iw.js?dpl=nogit-1792116633","/_next/static/chunks/3qiv_uhu50h9w.js?dpl=nogit-1792116633","/_next/static/chunks/1krfs1hu3nlgy.js?dpl=nogit-1792116633","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1792116633","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1792116633","/_next/static/chunks/3n19bhgn4l5zi.js?dpl=nogit-1792116633","/_next/static/chunks/3e1v2g1f52jrg.js?dpl=nogit-1792116633"],"UserProgressProvider"] 35:I[196793,["/_next/static/chunks/2um_u-bxbfm5w.js?dpl=nogit-1792116633","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792116633","/_next/static/chunks/0z347qy8pf1j3.js?dpl=nogit-1792116633","/_next/static/chunks/02eggfg7vwd9q.js?dpl=nogit-1792116633","/_next/static/chunks/1gm-hv-3-fp7_.js?dpl=nogit-1792116633","/_next/static/chunks/2xa0kylife8l8.js?dpl=nogit-1792116633","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792116633","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792116633","/_next/static/chunks/0kyh6u8hfv06h.js?dpl=nogit-1792116633","/_next/static/chunks/1zirpvj720iz0.js?dpl=nogit-1792116633","/_next/static/chunks/0mt9yp27gb-6w.js?dpl=nogit-1792116633","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792116633","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792116633","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1792116633","/_next/static/chunks/1jo7uanvci4-r.js?dpl=nogit-1792116633","/_next/static/chunks/0papbjd-vx5q5.js?dpl=nogit-1792116633","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792116633","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792116633","/_next/static/chunks/23f2tb1apskc8.js?dpl=nogit-1792116633","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792116633","/_next/static/chunks/3x5t_ry1ziv7r.js?dpl=nogit-1792116633","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792116633","/_next/static/chunks/0iapw94rt37n4.js?dpl=nogit-1792116633","/_next/static/chunks/2rvalcou1gw7v.js?dpl=nogit-1792116633","/_next/static/chunks/1c-rbkxmrcq7-.js?dpl=nogit-1792116633","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792116633","/_next/static/chunks/04wg1hnm8knmc.js?dpl=nogit-1792116633","/_next/static/chunks/3_six85qevnm3.js?dpl=nogit-1792116633","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792116633","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792116633","/_next/static/chunks/1vovlfobo0rf7.js?dpl=nogit-1792116633","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792116633","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1792116633","/_next/static/chunks/1jxfzuqocprar.js?dpl=nogit-1792116633","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792116633","/_next/static/chunks/3bplvoggfmxkd.js?dpl=nogit-1792116633","/_next/static/chunks/1drwxxqzvnao9.js?dpl=nogit-1792116633","/_next/static/chunks/0ltoscgl22itf.js?dpl=nogit-1792116633","/_next/static/chunks/442ru-jd_6vlx.js?dpl=nogit-1792116633","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1792116633","/_next/static/chunks/1zo1304bjzjl_.js?dpl=nogit-1792116633","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1792116633","/_next/static/chunks/0beaxlbowx15e.js?dpl=nogit-1792116633","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1792116633","/_next/static/chunks/2mbz851h6l6ul.js?dpl=nogit-1792116633","/_next/static/chunks/2yss1g0cimuwe.js?dpl=nogit-1792116633","/_next/static/chunks/0ioq6d8hla7h0.js?dpl=nogit-1792116633","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792116633","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792116633","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792116633","/_next/static/chunks/421rwkxv431i9.js?dpl=nogit-1792116633","/_next/static/chunks/2l63yczi1z2qz.js?dpl=nogit-1792116633","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1792116633","/_next/static/chunks/33y-ad8441jub.js?dpl=nogit-1792116633","/_next/static/chunks/1iyn56mfdoj9b.js?dpl=nogit-1792116633","/_next/static/chunks/1d4d88jx43_iw.js?dpl=nogit-1792116633","/_next/static/chunks/3qiv_uhu50h9w.js?dpl=nogit-1792116633","/_next/static/chunks/1krfs1hu3nlgy.js?dpl=nogit-1792116633","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1792116633","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1792116633","/_next/static/chunks/3n19bhgn4l5zi.js?dpl=nogit-1792116633","/_next/static/chunks/3e1v2g1f52jrg.js?dpl=nogit-1792116633"],"Gingerbread"] 37:I[716297,["/_next/static/chunks/2um_u-bxbfm5w.js?dpl=nogit-1792116633","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792116633","/_next/static/chunks/0z347qy8pf1j3.js?dpl=nogit-1792116633","/_next/static/chunks/02eggfg7vwd9q.js?dpl=nogit-1792116633","/_next/static/chunks/1gm-hv-3-fp7_.js?dpl=nogit-1792116633","/_next/static/chunks/2xa0kylife8l8.js?dpl=nogit-1792116633","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792116633","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792116633","/_next/static/chunks/0kyh6u8hfv06h.js?dpl=nogit-1792116633","/_next/static/chunks/1zirpvj720iz0.js?dpl=nogit-1792116633","/_next/static/chunks/0mt9yp27gb-6w.js?dpl=nogit-1792116633","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792116633","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792116633","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1792116633","/_next/static/chunks/1jo7uanvci4-r.js?dpl=nogit-1792116633","/_next/static/chunks/0papbjd-vx5q5.js?dpl=nogit-1792116633","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792116633","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792116633","/_next/static/chunks/23f2tb1apskc8.js?dpl=nogit-1792116633","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792116633","/_next/static/chunks/3x5t_ry1ziv7r.js?dpl=nogit-1792116633","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792116633","/_next/static/chunks/0iapw94rt37n4.js?dpl=nogit-1792116633","/_next/static/chunks/2rvalcou1gw7v.js?dpl=nogit-1792116633","/_next/static/chunks/1c-rbkxmrcq7-.js?dpl=nogit-1792116633","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792116633","/_next/static/chunks/04wg1hnm8knmc.js?dpl=nogit-1792116633","/_next/static/chunks/3_six85qevnm3.js?dpl=nogit-1792116633","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792116633","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792116633","/_next/static/chunks/1vovlfobo0rf7.js?dpl=nogit-1792116633","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792116633","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1792116633","/_next/static/chunks/1jxfzuqocprar.js?dpl=nogit-1792116633","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792116633","/_next/static/chunks/3bplvoggfmxkd.js?dpl=nogit-1792116633","/_next/static/chunks/1drwxxqzvnao9.js?dpl=nogit-1792116633","/_next/static/chunks/0ltoscgl22itf.js?dpl=nogit-1792116633","/_next/static/chunks/442ru-jd_6vlx.js?dpl=nogit-1792116633","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1792116633","/_next/static/chunks/1zo1304bjzjl_.js?dpl=nogit-1792116633","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1792116633","/_next/static/chunks/0beaxlbowx15e.js?dpl=nogit-1792116633","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1792116633","/_next/static/chunks/2mbz851h6l6ul.js?dpl=nogit-1792116633","/_next/static/chunks/2yss1g0cimuwe.js?dpl=nogit-1792116633","/_next/static/chunks/0ioq6d8hla7h0.js?dpl=nogit-1792116633","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792116633","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792116633","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792116633","/_next/static/chunks/421rwkxv431i9.js?dpl=nogit-1792116633","/_next/static/chunks/2l63yczi1z2qz.js?dpl=nogit-1792116633","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1792116633","/_next/static/chunks/33y-ad8441jub.js?dpl=nogit-1792116633","/_next/static/chunks/1iyn56mfdoj9b.js?dpl=nogit-1792116633","/_next/static/chunks/1d4d88jx43_iw.js?dpl=nogit-1792116633","/_next/static/chunks/3qiv_uhu50h9w.js?dpl=nogit-1792116633","/_next/static/chunks/1krfs1hu3nlgy.js?dpl=nogit-1792116633","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1792116633","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1792116633","/_next/static/chunks/3n19bhgn4l5zi.js?dpl=nogit-1792116633","/_next/static/chunks/3e1v2g1f52jrg.js?dpl=nogit-1792116633"],"TopicRecentlyViewed"] 38:I[647935,["/_next/static/chunks/2um_u-bxbfm5w.js?dpl=nogit-1792116633","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792116633","/_next/static/chunks/0z347qy8pf1j3.js?dpl=nogit-1792116633","/_next/static/chunks/02eggfg7vwd9q.js?dpl=nogit-1792116633","/_next/static/chunks/1gm-hv-3-fp7_.js?dpl=nogit-1792116633","/_next/static/chunks/2xa0kylife8l8.js?dpl=nogit-1792116633","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792116633","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792116633","/_next/static/chunks/0kyh6u8hfv06h.js?dpl=nogit-1792116633","/_next/static/chunks/1zirpvj720iz0.js?dpl=nogit-1792116633","/_next/static/chunks/0mt9yp27gb-6w.js?dpl=nogit-1792116633","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792116633","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792116633","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1792116633","/_next/static/chunks/1jo7uanvci4-r.js?dpl=nogit-1792116633","/_next/static/chunks/0papbjd-vx5q5.js?dpl=nogit-1792116633","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792116633","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792116633","/_next/static/chunks/23f2tb1apskc8.js?dpl=nogit-1792116633","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792116633","/_next/static/chunks/3x5t_ry1ziv7r.js?dpl=nogit-1792116633","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792116633","/_next/static/chunks/0iapw94rt37n4.js?dpl=nogit-1792116633","/_next/static/chunks/2rvalcou1gw7v.js?dpl=nogit-1792116633","/_next/static/chunks/1c-rbkxmrcq7-.js?dpl=nogit-1792116633","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792116633","/_next/static/chunks/04wg1hnm8knmc.js?dpl=nogit-1792116633","/_next/static/chunks/3_six85qevnm3.js?dpl=nogit-1792116633","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792116633","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792116633","/_next/static/chunks/1vovlfobo0rf7.js?dpl=nogit-1792116633","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792116633","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1792116633","/_next/static/chunks/1jxfzuqocprar.js?dpl=nogit-1792116633","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792116633","/_next/static/chunks/3bplvoggfmxkd.js?dpl=nogit-1792116633","/_next/static/chunks/1drwxxqzvnao9.js?dpl=nogit-1792116633","/_next/static/chunks/0ltoscgl22itf.js?dpl=nogit-1792116633","/_next/static/chunks/442ru-jd_6vlx.js?dpl=nogit-1792116633","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1792116633","/_next/static/chunks/1zo1304bjzjl_.js?dpl=nogit-1792116633","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1792116633","/_next/static/chunks/0beaxlbowx15e.js?dpl=nogit-1792116633","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1792116633","/_next/static/chunks/2mbz851h6l6ul.js?dpl=nogit-1792116633","/_next/static/chunks/2yss1g0cimuwe.js?dpl=nogit-1792116633","/_next/static/chunks/0ioq6d8hla7h0.js?dpl=nogit-1792116633","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792116633","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792116633","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792116633","/_next/static/chunks/421rwkxv431i9.js?dpl=nogit-1792116633","/_next/static/chunks/2l63yczi1z2qz.js?dpl=nogit-1792116633","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1792116633","/_next/static/chunks/33y-ad8441jub.js?dpl=nogit-1792116633","/_next/static/chunks/1iyn56mfdoj9b.js?dpl=nogit-1792116633","/_next/static/chunks/1d4d88jx43_iw.js?dpl=nogit-1792116633","/_next/static/chunks/3qiv_uhu50h9w.js?dpl=nogit-1792116633","/_next/static/chunks/1krfs1hu3nlgy.js?dpl=nogit-1792116633","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1792116633","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1792116633","/_next/static/chunks/3n19bhgn4l5zi.js?dpl=nogit-1792116633","/_next/static/chunks/3e1v2g1f52jrg.js?dpl=nogit-1792116633"],"RecordResourceResume"] 39:I[426249,["/_next/static/chunks/2um_u-bxbfm5w.js?dpl=nogit-1792116633","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792116633","/_next/static/chunks/0z347qy8pf1j3.js?dpl=nogit-1792116633","/_next/static/chunks/02eggfg7vwd9q.js?dpl=nogit-1792116633","/_next/static/chunks/1gm-hv-3-fp7_.js?dpl=nogit-1792116633","/_next/static/chunks/2xa0kylife8l8.js?dpl=nogit-1792116633","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792116633","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792116633","/_next/static/chunks/0kyh6u8hfv06h.js?dpl=nogit-1792116633","/_next/static/chunks/1zirpvj720iz0.js?dpl=nogit-1792116633","/_next/static/chunks/0mt9yp27gb-6w.js?dpl=nogit-1792116633","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792116633","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792116633","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1792116633","/_next/static/chunks/1jo7uanvci4-r.js?dpl=nogit-1792116633","/_next/static/chunks/0papbjd-vx5q5.js?dpl=nogit-1792116633","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792116633","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792116633","/_next/static/chunks/23f2tb1apskc8.js?dpl=nogit-1792116633","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792116633","/_next/static/chunks/3x5t_ry1ziv7r.js?dpl=nogit-1792116633","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792116633","/_next/static/chunks/0iapw94rt37n4.js?dpl=nogit-1792116633","/_next/static/chunks/2rvalcou1gw7v.js?dpl=nogit-1792116633","/_next/static/chunks/1c-rbkxmrcq7-.js?dpl=nogit-1792116633","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792116633","/_next/static/chunks/04wg1hnm8knmc.js?dpl=nogit-1792116633","/_next/static/chunks/3_six85qevnm3.js?dpl=nogit-1792116633","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792116633","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792116633","/_next/static/chunks/1vovlfobo0rf7.js?dpl=nogit-1792116633","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792116633","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1792116633","/_next/static/chunks/1jxfzuqocprar.js?dpl=nogit-1792116633","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792116633","/_next/static/chunks/3bplvoggfmxkd.js?dpl=nogit-1792116633","/_next/static/chunks/1drwxxqzvnao9.js?dpl=nogit-1792116633","/_next/static/chunks/0ltoscgl22itf.js?dpl=nogit-1792116633","/_next/static/chunks/442ru-jd_6vlx.js?dpl=nogit-1792116633","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1792116633","/_next/static/chunks/1zo1304bjzjl_.js?dpl=nogit-1792116633","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1792116633","/_next/static/chunks/0beaxlbowx15e.js?dpl=nogit-1792116633","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1792116633","/_next/static/chunks/2mbz851h6l6ul.js?dpl=nogit-1792116633","/_next/static/chunks/2yss1g0cimuwe.js?dpl=nogit-1792116633","/_next/static/chunks/0ioq6d8hla7h0.js?dpl=nogit-1792116633","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792116633","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792116633","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792116633","/_next/static/chunks/421rwkxv431i9.js?dpl=nogit-1792116633","/_next/static/chunks/2l63yczi1z2qz.js?dpl=nogit-1792116633","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1792116633","/_next/static/chunks/33y-ad8441jub.js?dpl=nogit-1792116633","/_next/static/chunks/1iyn56mfdoj9b.js?dpl=nogit-1792116633","/_next/static/chunks/1d4d88jx43_iw.js?dpl=nogit-1792116633","/_next/static/chunks/3qiv_uhu50h9w.js?dpl=nogit-1792116633","/_next/static/chunks/1krfs1hu3nlgy.js?dpl=nogit-1792116633","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1792116633","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1792116633","/_next/static/chunks/3n19bhgn4l5zi.js?dpl=nogit-1792116633","/_next/static/chunks/3e1v2g1f52jrg.js?dpl=nogit-1792116633"],"TopicQuestionBank"] 25:["$","$L34",null,{"bundle":{"topicIds":["50eb6c16-3d55-4bb9-bf41-19d5c925affb"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L35",null,{"feature":"question-bank"}],"$L36",["$","$L37",null,{"subjectCode":"a_level_maths__ocr_mei","subjectName":"Maths","topicSlug":"integrate-kx-n","topicTitle":"1.9.20 Integrate kx^n","sectionTitle":null,"activity":"worksheet","resourceTitle":"1.9.20 Integrate kx^n Questions","resourceHref":"/a-level/maths/ocr/mei/integrate-kx-n/questions/q1","saveTrail":false}],["$","$L38",null,{"href":"/a-level/maths/ocr/mei/integrate-kx-n/questions/q1","label":"1.9.20 Integrate kx^n","subjectCode":"a_level_maths__ocr_mei","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 32 – 1.9.20 Integrate kx^n - A Level Maths Questions\",\"description\":\"Practise 1.9.20 Integrate kx^n A Level Maths exam-style questions with worked solutions on MathsGenie.\",\"learningResourceType\":\"Practice questions\",\"educationalLevel\":\"A Level\",\"about\":{\"@type\":\"Thing\",\"name\":\"1.9.20 Integrate kx^n\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/ocr/mei/integrate-kx-n/questions/q32\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L39",null,{"backHref":"/a-level/maths/ocr/mei/questions","basePath":"/a-level/maths/ocr/mei/integrate-kx-n/questions","formulaUrl":null,"initialIndex":31,"initialQuestionWindow":[{"id":"3bf1d35e-bb20-4156-b36a-eb0c6ff57690","revisionId":"75152d25-4912-4e6e-983a-aea17e1451e6","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"easy","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"3a70568e-21e0-455a-ba46-fd3e0f50a1f0","label":null,"content":"Find \\[[\\int \\left( \\frac{4}{3}x^7 - 21\\sqrt[4]{x^3} + \\frac{5}{8x^6} \\right) \\mathrm{d}x\\]] giving your answer in simplest form.","metadata":null,"markscheme":"- Rewrites the integrand in index form as $\\frac{4}{3}x^7 - 21x^{\\frac{3}{4}} + \\frac{5}{8}x^{-6}$ B1\n- Applies the power rule for integration $\\int x^n \\mathrm{d}x = \\frac{x^{n+1}}{n+1}$ to at least two terms M1\n- Obtains any two of the simplified terms $\\frac{1}{6}x^8$, $-12x^{\\frac{7}{4}}$, or $-\\frac{1}{8x^5}$ correctly A1\n- Provides the fully simplified antiderivative $\\frac{1}{6}x^8 - 12x^{\\frac{7}{4}} - \\frac{1}{8x^5} + C$ including the constant of integration A1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"direct_integration","label":"Direct Term-by-Term Integration","steps":[{"type":"text","order":0,"title":"Convert to index form","content":"Before integrating, we need to rewrite each term of the integrand in the index form $ax^n$:\n\n* The first term is already in index form: \n $$\\frac{4}{3}x^7$$\n* The second term involves a radical. Recall that $\\sqrt[q]{x^p} = x^{\\frac{p}{q}}$, so:\n $$-21\\sqrt[4]{x^3} = -21x^{\\frac{3}{4}}$$\n* The third term has $x$ in the denominator. Recall that $\\frac{1}{x^n} = x^{-n}$, so:\n $$\\frac{5}{8x^6} = \\frac{5}{8}x^{-6}$$\n\nPutting this all together, the integral becomes:\n$$\\int \\left( \\frac{4}{3}x^7 - 21x^{\\frac{3}{4}} + \\frac{5}{8}x^{-6} \\right) \\mathrm{d}x$$"},{"type":"text","order":1,"title":"Recall the power rule","content":"Now we can apply the power rule of integration term-by-term. \n\nRecall that for any power $n \\neq -1$:\n$$\\int x^n \\mathrm{d}x = \\frac{x^{n+1}}{n+1} + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$I = \\int \\left( \\frac{4}{3}x^7 - 21x^{\\frac{3}{4}} + \\frac{5}{8}x^{-6} \\right) \\mathrm{d}x$"},{"latex":"$$I = \\htmlData{anchor=term1}{\\color{@sky}{\\frac{4}{3} \\cdot \\frac{x^8}{8}}} - \\htmlData{anchor=term2}{\\color{@amber}{21 \\cdot \\frac{x^{\\frac{7}{4}}}{\\frac{7}{4}}}} + \\htmlData{anchor=term3}{\\color{@purple}{\\frac{5}{8} \\cdot \\frac{x^{-5}}{-5}}} + C$"}],"highlights":[{"color":"sky","style":"box","anchor":"term1"},{"color":"amber","style":"box","anchor":"term2"},{"color":"purple","style":"box","anchor":"term3"}],"connections":[]},"order":2,"title":"Integrate term-by-term","description":"Apply the power rule, adding $1$ to the exponent and dividing by the new exponent for each term, and don't forget to add the constant of integration $C$."},{"type":"text","order":3,"title":"Simplify each term","content":"Let's simplify the coefficients for each of our terms:\n\n* **First term:**\n $$\\frac{4}{3} \\cdot \\frac{x^8}{8} = \\frac{4}{24}x^8 = \\frac{1}{6}x^8$$\n\n* **Second term:**\n $$-21 \\cdot \\frac{x^{\\frac{7}{4}}}{\\frac{7}{4}} = -21 \\cdot \\frac{4}{7}x^{\\frac{7}{4}} = -3 \\cdot 4x^{\\frac{7}{4}} = -12x^{\\frac{7}{4}}$$\n\n* **Third term:**\n $$\\frac{5}{8} \\cdot \\frac{x^{-5}}{-5} = -\\frac{5}{40}x^{-5} = -\\frac{1}{8}x^{-5} = -\\frac{1}{8x^5}$$"},{"type":"text","order":4,"title":"Write the final answer","content":"Combining all the simplified terms and adding the constant of integration $C$, we obtain the final answer:\n\n$$\\frac{1}{6}x^8 - 12x^{\\frac{7}{4}} - \\frac{1}{8x^5} + C$$"}]}],"generatedAt":"2026-06-09T04:23:38.690Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"3a70568e-21e0-455a-ba46-fd3e0f50a1f0:direct_integration","label":"Direct Term-by-Term Integration","steps":[{"type":"text","order":0,"title":"Convert to index form","content":"Before integrating, we need to rewrite each term of the integrand in the index form $ax^n$:\n\n* The first term is already in index form: \n $$\\frac{4}{3}x^7$$\n* The second term involves a radical. Recall that $\\sqrt[q]{x^p} = x^{\\frac{p}{q}}$, so:\n $$-21\\sqrt[4]{x^3} = -21x^{\\frac{3}{4}}$$\n* The third term has $x$ in the denominator. Recall that $\\frac{1}{x^n} = x^{-n}$, so:\n $$\\frac{5}{8x^6} = \\frac{5}{8}x^{-6}$$\n\nPutting this all together, the integral becomes:\n$$\\int \\left( \\frac{4}{3}x^7 - 21x^{\\frac{3}{4}} + \\frac{5}{8}x^{-6} \\right) \\mathrm{d}x$$"},{"type":"text","order":1,"title":"Recall the power rule","content":"Now we can apply the power rule of integration term-by-term. \n\nRecall that for any power $n \\neq -1$:\n$$\\int x^n \\mathrm{d}x = \\frac{x^{n+1}}{n+1} + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$I = \\int \\left( \\frac{4}{3}x^7 - 21x^{\\frac{3}{4}} + \\frac{5}{8}x^{-6} \\right) \\mathrm{d}x$"},{"latex":"$$I = \\htmlData{anchor=term1}{\\color{@sky}{\\frac{4}{3} \\cdot \\frac{x^8}{8}}} - \\htmlData{anchor=term2}{\\color{@amber}{21 \\cdot \\frac{x^{\\frac{7}{4}}}{\\frac{7}{4}}}} + \\htmlData{anchor=term3}{\\color{@purple}{\\frac{5}{8} \\cdot \\frac{x^{-5}}{-5}}} + C$"}],"highlights":[{"color":"sky","style":"box","anchor":"term1"},{"color":"amber","style":"box","anchor":"term2"},{"color":"purple","style":"box","anchor":"term3"}],"connections":[]},"order":2,"title":"Integrate term-by-term","description":"Apply the power rule, adding $1$ to the exponent and dividing by the new exponent for each term, and don't forget to add the constant of integration $C$."},{"type":"text","order":3,"title":"Simplify each term","content":"Let's simplify the coefficients for each of our terms:\n\n* **First term:**\n $$\\frac{4}{3} \\cdot \\frac{x^8}{8} = \\frac{4}{24}x^8 = \\frac{1}{6}x^8$$\n\n* **Second term:**\n $$-21 \\cdot \\frac{x^{\\frac{7}{4}}}{\\frac{7}{4}} = -21 \\cdot \\frac{4}{7}x^{\\frac{7}{4}} = -3 \\cdot 4x^{\\frac{7}{4}} = -12x^{\\frac{7}{4}}$$\n\n* **Third term:**\n $$\\frac{5}{8} \\cdot \\frac{x^{-5}}{-5} = -\\frac{5}{40}x^{-5} = -\\frac{1}{8}x^{-5} = -\\frac{1}{8x^5}$$"},{"type":"text","order":4,"title":"Write the final answer","content":"Combining all the simplified terms and adding the constant of integration $C$, we obtain the final answer:\n\n$$\\frac{1}{6}x^8 - 12x^{\\frac{7}{4}} - \\frac{1}{8x^5} + C$$"}]}],"generatedAt":"2026-06-09T04:23:52.685Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"3c78da78-8789-47c6-811d-8f21988df751","revisionId":"eb065649-66bf-403e-9e65-ab818abe5f77","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"easy","tier":null,"totalMarks":2,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"c64e692a-9061-4fc1-b4f3-faed23396016","label":null,"content":"Find\n\\[\\int (x^3 + x^{\\frac{1}{3}}) \\, dx\\]","metadata":null,"markscheme":"- Integrates at least one term correctly to obtain $\\frac{x^4}{4}$ or $\\frac{3}{4}x^{\\frac{4}{3}}$ M1\n- Obtains the full correct expression $\\frac{x^4}{4} + \\frac{3x^{\\frac{4}{3}}}{4} + C$ A1","marks":2,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"direct_integration","label":"Direct Integration","steps":[{"type":"text","order":0,"title":"Identify the method","content":"To solve this indefinite integral, we can apply **Direct Integration** by integrating each term separately using the power rule of integration.\n\nThe power rule states that for any real constant $n \\neq -1$:\n\n$$\\int x^n \\, dx = \\frac{x^{n+1}}{n+1} + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^{\\htmlData{anchor=power1}{\\color{@sky}{3}}}$"},{"latex":"$$\\int x^{\\color{@sky}{3}} \\, dx = \\frac{x^{\\color{@sky}{3}+1}}{\\color{@sky}{3}+1} = \\htmlData{anchor=result1}{\\color{@sky}{\\frac{x^4}{4}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"power1"},{"color":"sky","style":"text-color","anchor":"result1"}],"connections":[{"to":"result1","from":"power1","color":"sky","label":"integrated","style":"solid"}]},"order":1,"title":"Integrate the first term","description":"First, let's apply the rule to the term $x^3$. Here, the exponent is $n = 3$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^{\\htmlData{anchor=power2}{\\color{@amber}{\\frac{1}{3}}}}$"},{"latex":"$$\\int x^{\\color{@amber}{\\frac{1}{3}}} \\, dx = \\frac{x^{\\color{@amber}{\\frac{1}{3}}+1}}{\\color{@amber}{\\frac{1}{3}}+1} = \\frac{x^{\\frac{4}{3}}}{\\frac{4}{3}} = \\htmlData{anchor=result2}{\\color{@amber}{\\frac{3}{4}x^{\\frac{4}{3}}}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"power2"},{"color":"amber","style":"text-color","anchor":"result2"}],"connections":[{"to":"result2","from":"power2","color":"amber","label":"integrated","style":"solid"}]},"order":2,"title":"Integrate the second term","description":"Next, apply the power rule to the term $x^{\\frac{1}{3}}$ where $n = \\frac{1}{3}$. Since $\\frac{1}{3} + 1 = \\frac{4}{3}$, dividing by $\\frac{4}{3}$ is the same as multiplying by $\\frac{3}{4}$."},{"type":"text","order":3,"title":"Combine the results","content":"Now, we combine the two integrated terms and add the constant of integration, $C$, to get our final answer:\n\n$$\\int (x^3 + x^{\\frac{1}{3}}) \\, dx = \\frac{x^4}{4} + \\frac{3}{4}x^{\\frac{4}{3}} + C$$"}]}],"generatedAt":"2026-06-09T04:28:12.003Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"c64e692a-9061-4fc1-b4f3-faed23396016:direct_integration","label":"Direct Integration","steps":[{"type":"text","order":0,"title":"Identify the method","content":"To solve this indefinite integral, we can apply **Direct Integration** by integrating each term separately using the power rule of integration.\n\nThe power rule states that for any real constant $n \\neq -1$:\n\n$$\\int x^n \\, dx = \\frac{x^{n+1}}{n+1} + C$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^{\\htmlData{anchor=power1}{\\color{@sky}{3}}}$"},{"latex":"$$\\int x^{\\color{@sky}{3}} \\, dx = \\frac{x^{\\color{@sky}{3}+1}}{\\color{@sky}{3}+1} = \\htmlData{anchor=result1}{\\color{@sky}{\\frac{x^4}{4}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"power1"},{"color":"sky","style":"text-color","anchor":"result1"}],"connections":[{"to":"result1","from":"power1","color":"sky","label":"integrated","style":"solid"}]},"order":1,"title":"Integrate the first term","description":"First, let's apply the rule to the term $x^3$. Here, the exponent is $n = 3$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^{\\htmlData{anchor=power2}{\\color{@amber}{\\frac{1}{3}}}}$"},{"latex":"$$\\int x^{\\color{@amber}{\\frac{1}{3}}} \\, dx = \\frac{x^{\\color{@amber}{\\frac{1}{3}}+1}}{\\color{@amber}{\\frac{1}{3}}+1} = \\frac{x^{\\frac{4}{3}}}{\\frac{4}{3}} = \\htmlData{anchor=result2}{\\color{@amber}{\\frac{3}{4}x^{\\frac{4}{3}}}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"power2"},{"color":"amber","style":"text-color","anchor":"result2"}],"connections":[{"to":"result2","from":"power2","color":"amber","label":"integrated","style":"solid"}]},"order":2,"title":"Integrate the second term","description":"Next, apply the power rule to the term $x^{\\frac{1}{3}}$ where $n = \\frac{1}{3}$. Since $\\frac{1}{3} + 1 = \\frac{4}{3}$, dividing by $\\frac{4}{3}$ is the same as multiplying by $\\frac{3}{4}$."},{"type":"text","order":3,"title":"Combine the results","content":"Now, we combine the two integrated terms and add the constant of integration, $C$, to get our final answer:\n\n$$\\int (x^3 + x^{\\frac{1}{3}}) \\, dx = \\frac{x^4}{4} + \\frac{3}{4}x^{\\frac{4}{3}} + C$$"}]}],"generatedAt":"2026-06-09T04:28:21.776Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"3fa00e4a-bd26-4198-bcb7-2cc4b5d846c4","revisionId":"9c9b6455-d37a-480d-aead-7cf05a33a801","questionNumber":null,"questionType":"LA","specification":"A researcher determines that the rate of expansion of a specialized polymer, $V$ mm$^3$, with respect to time $t$ seconds, is modeled by the equation:\n\n\\[ \\frac{dV}{dt} = 30t^5 + 16t^3 - \\frac{14}{t^8}, \\quad t > 0 \\]","difficultyBand":"easy","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"0ea4dd1a-70b5-4a1d-9526-c346b9ae18b5","label":null,"content":"Find a general expression for $V$ in terms of $t$, giving your answer in its simplest form.","metadata":null,"markscheme":"- Rewrites the reciprocal term using a negative index as $14t^{-8}$ M1\n- Integrates the expression, increasing at least two powers by 1 M1\n- Correctly integrates the first two terms to $5t^6 + 4t^4$ A1\n- Correctly integrates the final term and includes the constant of integration to give $5t^6 + 4t^4 + \\frac{2}{t^7} + C$ A1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"direct_integration","label":"Direct Integration","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"To find $V$ from the rate of change $\\frac{dV}{dt}$, we need to integrate the rate of change equation with respect to $t$:\\n\\n$$V = \\int \\left(30t^5 + 16t^3 - \\frac{14}{t^8}\\right) dt$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"Before integrating, rewrite the fractional term using a negative exponent so we can easily apply the power rule of integration:"},{"latex":"$$-\\frac{14}{\\htmlData{anchor=t-denom}{t^8}} = -14\\htmlData{anchor=t-neg}{t^{-8}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"t-denom"},{"color":"sky","style":"box","anchor":"t-neg"}],"connections":[{"to":"t-neg","from":"t-denom","color":"sky","label":"negative index","style":"solid"}]},"order":1,"title":"Rewrite the fraction term","description":"This step earns the first method mark (M1) by converting the reciprocal term into index form."},{"type":"text","order":2,"title":"Apply the power rule","content":"We now integrate term-by-term using the fundamental power rule of integration:\\n\\n$$\\int t^n dt = \\frac{t^{n+1}}{n+1} \\quad \\text{for } n \\neq -1$$\\n\\nLet's apply this to each term of our rewritten expression:\\n\\n$$\\int \\left(30t^5 + 16t^3 - 14t^{-8}\\right) dt$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"For the first two polynomial terms, increase the power by 1 and divide by the new power:"},{"latex":"$$\\int \\htmlData{anchor=term1}{30t^5} dt = 30 \\left(\\frac{t^6}{6}\\right) = \\htmlData{anchor=res1}{\\color{@green}{5t^6}}$"},{"latex":"$$\\int \\htmlData{anchor=term2}{16t^3} dt = 16 \\left(\\frac{t^4}{4}\\right) = \\htmlData{anchor=res2}{\\color{@teal}{4t^4}}$"}],"highlights":[{"color":"green","style":"underline","anchor":"term1"},{"color":"green","style":"text-color","anchor":"res1"},{"color":"teal","style":"underline","anchor":"term2"},{"color":"teal","style":"text-color","anchor":"res2"}],"connections":[{"to":"res1","from":"term1","color":"green","label":null,"style":"solid"},{"to":"res2","from":"term2","color":"teal","label":null,"style":"solid"}]},"order":3,"title":"Integrate the first two terms","description":"Increasing the powers by 1 earns a method mark (M1). Simplifying these first two terms correctly to $5t^6 + 4t^4$ earns an accuracy mark (A1)."},{"type":"hyper_latex","block":{"nodes":[{"text":"Apply the same rule to the term with the negative exponent:"},{"latex":"$$\\int \\htmlData{anchor=negterm}{-14t^{-8}} dt = -14 \\left( \\frac{t^{-7}}{-7} \\right) = \\htmlData{anchor=negres}{\\color{@purple}{2t^{-7}}}$"},{"text":"This can be written back in reciprocal form as:"},{"latex":"$$\\color{@purple}{\\frac{2}{t^7}}$"}],"highlights":[{"color":"purple","style":"underline","anchor":"negterm"},{"color":"purple","style":"text-color","anchor":"negres"}],"connections":[{"to":"negres","from":"negterm","color":"purple","label":null,"style":"solid"}]},"order":4,"title":"Integrate negative term","description":"Be careful with negative signs when dividing: $-14$ divided by $-7$ becomes $+2$."},{"type":"text","order":5,"title":"State the final expression","content":"Combine all of the integrated terms and remember to include the constant of integration, $C$:\\n\\n$$V = 5t^6 + 4t^4 + \\frac{2}{t^7} + C$$\\n\\nThis completes our general expression for $V$ and earns the final accuracy mark (**A1**)."}]}],"generatedAt":"2026-06-09T04:38:18.497Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"0ea4dd1a-70b5-4a1d-9526-c346b9ae18b5:direct_integration","label":"Direct Integration","steps":[{"type":"text","order":0,"title":"Set up the integral","content":"To find $V$ from the rate of change $\\frac{dV}{dt}$, we need to integrate the rate of change equation with respect to $t$:\\n\\n$$V = \\int \\left(30t^5 + 16t^3 - \\frac{14}{t^8}\\right) dt$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"Before integrating, rewrite the fractional term using a negative exponent so we can easily apply the power rule of integration:"},{"latex":"$$-\\frac{14}{\\htmlData{anchor=t-denom}{t^8}} = -14\\htmlData{anchor=t-neg}{t^{-8}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"t-denom"},{"color":"sky","style":"box","anchor":"t-neg"}],"connections":[{"to":"t-neg","from":"t-denom","color":"sky","label":"negative index","style":"solid"}]},"order":1,"title":"Rewrite the fraction term","description":"This step earns the first method mark (M1) by converting the reciprocal term into index form."},{"type":"text","order":2,"title":"Apply the power rule","content":"We now integrate term-by-term using the fundamental power rule of integration:\\n\\n$$\\int t^n dt = \\frac{t^{n+1}}{n+1} \\quad \\text{for } n \\neq -1$$\\n\\nLet's apply this to each term of our rewritten expression:\\n\\n$$\\int \\left(30t^5 + 16t^3 - 14t^{-8}\\right) dt$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"For the first two polynomial terms, increase the power by 1 and divide by the new power:"},{"latex":"$$\\int \\htmlData{anchor=term1}{30t^5} dt = 30 \\left(\\frac{t^6}{6}\\right) = \\htmlData{anchor=res1}{\\color{@green}{5t^6}}$"},{"latex":"$$\\int \\htmlData{anchor=term2}{16t^3} dt = 16 \\left(\\frac{t^4}{4}\\right) = \\htmlData{anchor=res2}{\\color{@teal}{4t^4}}$"}],"highlights":[{"color":"green","style":"underline","anchor":"term1"},{"color":"green","style":"text-color","anchor":"res1"},{"color":"teal","style":"underline","anchor":"term2"},{"color":"teal","style":"text-color","anchor":"res2"}],"connections":[{"to":"res1","from":"term1","color":"green","label":null,"style":"solid"},{"to":"res2","from":"term2","color":"teal","label":null,"style":"solid"}]},"order":3,"title":"Integrate the first two terms","description":"Increasing the powers by 1 earns a method mark (M1). Simplifying these first two terms correctly to $5t^6 + 4t^4$ earns an accuracy mark (A1)."},{"type":"hyper_latex","block":{"nodes":[{"text":"Apply the same rule to the term with the negative exponent:"},{"latex":"$$\\int \\htmlData{anchor=negterm}{-14t^{-8}} dt = -14 \\left( \\frac{t^{-7}}{-7} \\right) = \\htmlData{anchor=negres}{\\color{@purple}{2t^{-7}}}$"},{"text":"This can be written back in reciprocal form as:"},{"latex":"$$\\color{@purple}{\\frac{2}{t^7}}$"}],"highlights":[{"color":"purple","style":"underline","anchor":"negterm"},{"color":"purple","style":"text-color","anchor":"negres"}],"connections":[{"to":"negres","from":"negterm","color":"purple","label":null,"style":"solid"}]},"order":4,"title":"Integrate negative term","description":"Be careful with negative signs when dividing: $-14$ divided by $-7$ becomes $+2$."},{"type":"text","order":5,"title":"State the final expression","content":"Combine all of the integrated terms and remember to include the constant of integration, $C$:\\n\\n$$V = 5t^6 + 4t^4 + \\frac{2}{t^7} + C$$\\n\\nThis completes our general expression for $V$ and earns the final accuracy mark (**A1**)."}]}],"generatedAt":"2026-06-09T04:38:27.990Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"a_level_maths__ocr_mei","summaries":[{"id":"19b1d184-7b2e-4eab-8e8e-e0f619c14e25","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"91ca954b-1bdb-47fa-96fa-5f9ac25fb5f7","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"9477a240-9ae6-4f66-bab4-556745c10e10","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"af51146c-eb8e-481b-b344-e5fc3a1c1086","difficultyBand":"easy","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"33eaf0c2-c8d6-4e1c-bb48-6c1ef08aac6d","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"3831be23-9d8f-4c4f-b16f-263d62d79627","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":false},{"id":"e7420bc2-5fd8-4afa-b008-b3f8d851a4bd","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"79c30fa9-a5dd-42d9-abf9-eab78bc78177","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"dc6d338f-b9f1-44d0-8427-04c5102e74d8","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"04ebdcb9-9fb6-4d98-a0cf-ce942c92ef1d","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"077f0048-b271-4300-8b0c-c4573895aabb","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"09a9bff7-3d5f-4ea7-86e9-e0ebdf77d8b1","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"09ea0b5c-19b3-487b-b3c8-9d30c7f0cb18","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"0afa7a3c-4e9a-488b-9ca3-5594de5450d2","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"0ebe6bbe-b6d2-4725-b255-a4dfaaf64e12","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"16e688aa-9193-46ee-901b-e45d1ebe0d42","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1b65118c-6ba7-4b39-92be-338ac0c8e13f","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1d419b9e-87a0-4fa4-94a4-c3a0d71d171c","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1fda6d6f-10fe-48e4-a618-c2ee9e39e51a","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2641fa14-1752-403c-bc7b-fdecb6f1b8d0","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":false},{"id":"2a1270c9-5bcf-47de-9be7-dc1979a5e630","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2a961770-ef5f-4ce4-b128-3eab2bfb3412","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2baf7366-c0ab-431f-9aa4-1b167daaf5c8","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"2f86eff4-299f-4e87-8b33-6124a9ae166d","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":false},{"id":"304561c8-bb51-4f14-8e2f-64bc58dd157a","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"32831ce7-3931-4f98-8c2e-827897ee25c4","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"36e54636-bda2-4f83-9e20-ac19d5c6a13a","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"38252659-3641-4dd0-81e7-9a6c7bffaf85","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"39740dc4-92d4-4d05-bf4c-11e2d764cf8b","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"3b0aae70-80a9-498c-aaad-6020ae06da80","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"3bf1d35e-bb20-4156-b36a-eb0c6ff57690","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"3c78da78-8789-47c6-811d-8f21988df751","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"3fa00e4a-bd26-4198-bcb7-2cc4b5d846c4","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"434679a2-8b43-470d-803b-7033609f2042","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"47a20a87-0fda-4a46-8d13-c48e57f5a16a","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"48824f7f-2a0f-47bd-97db-eddb1853dd8d","difficultyBand":"easy","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"5c6c1d8b-3534-4e84-92f2-1df06098207c","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"5d213950-75ff-4563-9b09-d0cda44eb7c8","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"5d7ac529-8e05-4e15-9892-da0787f34491","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"60ce1b08-e893-4ad5-b663-f310e32fd2ed","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"657d9a3c-372a-4e0a-b443-88e3d0b80ef0","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"69d24359-6717-4e72-90d7-2dc821419d50","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"6c4d2c9d-5cda-4039-bd2c-849ca673c305","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"71745279-355c-44b2-9959-758980be5644","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"718077b9-7f0c-4005-8c83-52f6284158ca","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"7255c9d2-71eb-4b9f-8b18-7b9efd299b7b","difficultyBand":"medium","calculatorUse":null,"totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"72641cd0-44c9-4ed5-990b-e0e220ef3b23","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"7615f8a8-c5bd-401a-aefd-1d08f0eb2130","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"8210c471-575c-4cc0-961a-1d68e442b90c","difficultyBand":"medium","calculatorUse":null,"totalMarks":11,"questionType":"LA","isCalculation":true},{"id":"860d0900-dbab-4bb9-86a0-a63271403772","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"8c61a1b3-184d-4cf1-8aaf-93d5bac10cec","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"90d1e92e-dc8f-4449-a7a7-b32949be4c32","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"911b39cd-28a0-4cb2-9f84-17b37e191056","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"92b0a1af-00ff-4561-9a48-961c1db57937","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"9b92f948-1837-43f2-8b18-373ba8d6ec3b","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":false},{"id":"a2e71239-e7e4-4657-a9c9-a06b2474aa3e","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"a5ea7b20-d15e-4f41-88bd-f074d8d1b741","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":false},{"id":"a9b4224f-432a-4549-8b38-ab486616e0c3","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"abf1e60f-93dd-4925-8b37-e47402be0402","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"b2402387-d8b0-4b28-bf6f-005bf2b9db52","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"b688b571-4ac7-487b-abc5-1d579afe913c","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"bdb49344-d742-4b70-98ee-1614c5065fd0","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"c04b972d-6b15-480b-8e07-e9f6bd237c83","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"cf4e92c2-3d88-4a82-a414-816eff21425c","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d19956f5-89c5-43a1-ae74-d8cda96e04d8","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d632eb7a-94f6-4466-b24f-e616849986fb","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d6adf4be-96a5-4bd3-a14d-6c1950733744","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"dd4288e9-243d-4d0c-9644-cd4ef9b77aa8","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"e1c2726f-f858-4398-9f7e-a1660f448139","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"e1eac852-1025-4ebc-b7cd-a017e7dd49ea","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"e20e0ef3-27e7-48ea-9653-88b936d67e6d","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"e261c190-ee40-4afc-9760-14e4ffb6e3a0","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e48218e3-ad9e-493c-a2d9-a4322a7d34ef","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"e50c3612-2184-4c24-94df-49d012e6f271","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e6bd135f-1e29-4f18-adcc-6c4a43f977c1","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"ecce763a-3485-4ba5-9462-0d3be15eb625","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"ed2bad36-0cde-40f5-a1db-ed7b4d9d7e47","difficultyBand":"medium","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"f85b0743-6706-4fee-8cd1-16965d1b2437","difficultyBand":"hard","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"fd56402b-2429-444d-8e40-e1993d46c168","difficultyBand":"hard","calculatorUse":null,"totalMarks":10,"questionType":"LA","isCalculation":true},{"id":"ff17d715-f4ef-474c-933c-ad3f89a0a51b","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false}],"title":"1.9.20 Integrate kx^n","topicId":"50eb6c16-3d55-4bb9-bf41-19d5c925affb","topicSidebar":"$L3a","topicSlug":"integrate-kx-n","topicTitle":"1.9.20 Integrate kx^n"}]}],"$L3b","$L3c"]}]

1.9.20 Integrate kx^n

Question 23

1.9.20 Integrate kx^n Questions

1.9.20 Integrate kx^n

Question 23

1.9.20 Integrate kx^n Questions