1cf6a1a-1cf3-4500-bd3e-a39b035b96ac","label":"0","content":"Find $\\int \\left( 10 - \\frac{6}{x^2} \\right) dx$ writing your answer in its simplest form.","markscheme":"$$$\\int 10 - 6x^{-2} dx = 10x + 6x^{-1} + C$$ 4","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Rewrites term as $6x^{-2}$.","code":"B"},{"id":"p2","marks":1,"description":"Integrates 10 to $10x$.","code":"M"},{"id":"p3","marks":1,"description":"Integrates $-6x^{-2}$ to $6x^{-1}$.","code":"A"},{"id":"p4","marks":1,"description":"Includes $+ C$.","code":"B"}],"finalAnswers":["10x + 6x^{-1} + C"],"generatedAt":"2026-05-12T02:14:06.620Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"power_rule","label":"Use the power rule","steps":[{"type":"text","order":0,"title":"Rewrite the fraction","content":"Rewrite the reciprocal term using a negative index:\n\n$$\\frac{6}{x^2}=6x^{-2}$$\n\nSo the integral becomes\n\n$$\\int\\left(10-6x^{-2}\\right)\\,dx.$$\n\nThis exposes the power of $x$, which is needed for the integration rule.","highlights":[{"text":"Find $\\int \\left( 10 - \\frac{6}{x^2} \\right) dx$","location":"specification"}]},{"type":"text","order":1,"title":"Recall the power rule","content":"For any power $n\\ne -1$,\n\n$$\\int x^n\\,dx=\\frac{x^{n+1}}{n+1}+C.$$\n\nIntegrate each term separately. Add the constant of integration $C$ once, at the end."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\int 10\\,dx = \\htmlData{anchor=constant}{\\color{@sky}{10x}}"},{"latex":"\\int -6x^{-2}\\,dx = \\htmlData{anchor=power}{\\color{@amber}{6x^{-1}}}"},{"latex":"\\int (10-6x^{-2})\\,dx = \\htmlData{anchor=answer}{\\color{@green}{10x+6x^{-1}+C}}"}],"highlights":[{"color":"sky","style":"text-color","anchor":"constant"},{"color":"amber","style":"text-color","anchor":"power"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"answer","from":"constant","color":"sky","label":"combine terms"},{"to":"answer","from":"power","color":"amber","label":"combine terms"}]},"order":2,"title":"Integrate each term","description":"Apply the power rule to the constant term and to the term with exponent $-2$."},{"type":"text","order":3,"title":"Write the final form","content":"Combine the integrated terms and include the arbitrary constant:\n\n$$\\int\\left(10-6x^{-2}\\right)\\,dx=10x+6x^{-1}+C.$$\n\nThis earns the marks for applying the power rule to both terms, simplifying the negative coefficient correctly, and including $C$. Since the question asks for the simplest form, you may also write\n\n$$\\boxed{10x+\\frac{6}{x}+C}.$$"}]}],"generatedAt":"2026-07-20T16:59:53.872Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"09a9bff7-3d5f-4ea7-86e9-e0ebdf77d8b1","revisionId":"274cf742-846a-4ae7-aa70-51d8b95282c2","questionNumber":null,"questionType":"LA","specification":"","difficulty":3,"difficultyBand":"easy","totalMarks":4,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"bdfb86fd-c9dd-49ae-8471-89aec52ad4a5","label":null,"content":"Find\n\n\\[\\int \\left( 12x^5 - \\frac{5}{3x^6} + 4 \\right) dx \\]\n\ngiving each term in simplest form.","markscheme":"- Attempts to integrate at least one term by increasing the power of $x$ by 1 M1\n- Integrates $12x^5$ correctly to obtain $2x^6$ A1\n- Integrates $-\\frac{5}{3}x^{-6}$ correctly to obtain $+\\frac{1}{3}x^{-5}$ or $+\\frac{1}{3x^5}$ A1\n- Integrates $4$ to obtain $4x$ and includes the constant of integration $+ c$ B1","marks":4,"order":0,"markingRubric":null,"explanation":{"methods":[{"id":"power_rule","label":"Term-by-Term Integration","steps":[{"type":"text","order":0,"title":"Rewrite with negative exponents","content":"To integrate term-by-term using the power rule, we first want to express all terms with $x$ as powers of $x$.\n\nWe rewrite the fraction term using a negative exponent:\n$$\\frac{5}{3x^6} = \\frac{5}{3}x^{-6}$$\n\nThis let

1.7 Integration

Question 16

1.7 Integration Questions

1.7 Integration

Question 16

1.7 Integration Questions