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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\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L3b",null,{"backHref":"/a-level/maths/ocr/mei/questions","basePath":"/a-level/maths/ocr/mei/integrate-kx-n/questions","formulaUrl":null,"initialIndex":31,"initialQuestionWindow":[{"id":"39740dc4-92d4-4d05-bf4c-11e2d764cf8b","revisionId":"03084860-32f7-4142-9808-1ff4334ab97c","questionNumber":null,"questionType":"LA","specification":"A research probe measures the rate of accumulation of cosmic dust, \$D\$, on a satellite's surface over time, \$t\$, in years. The rate is modelled by the function:\n\n\\[ \\frac{dD}{dt} = \\frac{5}{8}t^4 - \\frac{2}{7t^5} + \\sqrt{3} \\]","difficulty":3,"difficultyBand":"easy","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"60ba3f4c-f8db-4c70-8908-c05b3b163ee5","label":null,"content":"Find the general expression for \$D(t)\$ by evaluating:\n\n\\[ \\int \\left( \\frac{5}{8}t^4 - \\frac{2}{7t^5} + \\sqrt{3} \\right) dt \\]\n\nsimplifying your answer.","markscheme":"- Rewrites the second term in the integrand using a negative index: $-\\frac{2}{7}t^{-5}$ M1\n- Integrates the first term to get $\\frac{5}{8} \\times \\frac{t^5}{5} = \\frac{1}{8}t^5$ A1\n- Integrates the second term to get $-\\frac{2}{7} \\times \\frac{t^{-4}}{-4} = \\frac{1}{14}t^{-4}$ or $\\frac{1}{14t^4}$ A1\n- States the final simplified expression $\\frac{1}{8}t^5 + \\frac{1}{14t^4} + \\sqrt{3}t + c$ including the constant of integration A1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"power_rule","label":"Term-by-Term Integration","steps":[{"type":"text","order":0,"title":"Rewrite the terms","content":"To integrate this expression term-by-term, we first rewrite any rational terms with negative exponents.\n\nSpecifically, the second term $-\\frac{2}{7t^5}$ can be rewritten using the rule $x^{-n} = \\frac{1}{x^n}$:\n\n$$-\\frac{2}{7t^5} = -\\frac{2}{7}t^{-5}$$\n\nThis gives us the rewritten integral to evaluate:\n\n$$\\int \\left( \\frac{5}{8}t^4 - \\frac{2}{7}t^{-5} + \\sqrt{3} \\right) dt$$"},{"type":"text","order":1,"title":"Integrate the first term","content":"Now we integrate each term individually using the power rule for integration:\n\n$$\\int x^n dx = \\frac{x^{n+1}}{n+1} \\quad (n \\neq -1)$$\n\nFor the first term, $\\frac{5}{8}t^4$:\n\n$$\\int \\frac{5}{8}t^4 dt = \\frac{5}{8} \\times \\frac{t^{4+1}}{4+1} = \\frac{5}{8} \\times \\frac{t^5}{5} = \\frac{1}{8}t^5$$"},{"type":"text","order":2,"title":"Integrate the second term","content":"Next, we apply the power rule to the second term, $-\\frac{2}{7}t^{-5}$:\n\n$$\\int -\\frac{2}{7}t^{-5} dt = -\\frac{2}{7} \\times \\frac{t^{-5+1}}{-5+1} = -\\frac{2}{7} \\times \\frac{t^{-4}}{-4}$$\n\nMultiplying the fractions:\n\n$$\\left(-\\frac{2}{7}\\right) \\times \\left(\\frac{t^{-4}}{-4}\\right) = \\frac{2}{28}t^{-4} = \\frac{1}{14}t^{-4} = \\frac{1}{14t^4}$$"},{"type":"text","order":3,"title":"Integrate the constant term","content":"Finally, we integrate the constant term, $\\sqrt{3}$:\n\n$$\\int \\sqrt{3} dt = \\sqrt{3}t$$"},{"type":"text","order":4,"title":"Combine and simplify","content":"Combining the results from our term-by-term integration and remembering to include the constant of integration, $c$, we obtain the general expression for $D(t)$:\n\n$$D(t) = \\frac{1}{8}t^5 + \\frac{1}{14t^4} + \\sqrt{3}t + c$$"}]}],"generatedAt":"2026-06-09T04:46:19.754Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"60ba3f4c-f8db-4c70-8908-c05b3b163ee5:power_rule","label":"Term-by-Term Integration","steps":[{"type":"text","order":0,"title":"Rewrite the terms","content":"To integrate this expression term-by-term, we first rewrite any rational terms with negative exponents.\n\nSpecifically, the second term $-\\frac{2}{7t^5}$ can be rewritten using the rule $x^{-n} = \\frac{1}{x^n}$:\n\n$$-\\frac{2}{7t^5} = -\\frac{2}{7}t^{-5}$$\n\nThis gives us the rewritten integral to evaluate:\n\n$$\\int \\left( \\frac{5}{8}t^4 - \\frac{2}{7}t^{-5} + \\sqrt{3} \\right) dt$$"},{"type":"text","order":1,"title":"Integrate the first term","content":"Now we integrate each term individually using the power rule for integration:\n\n$$\\int x^n dx = \\frac{x^{n+1}}{n+1} \\quad (n \\neq -1)$$\n\nFor the first term, $\\frac{5}{8}t^4$:\n\n$$\\int \\frac{5}{8}t^4 dt = \\frac{5}{8} \\times \\frac{t^{4+1}}{4+1} = \\frac{5}{8} \\times \\frac{t^5}{5} = \\frac{1}{8}t^5$$"},{"type":"text","order":2,"title":"Integrate the second term","content":"Next, we apply the power rule to the second term, $-\\frac{2}{7}t^{-5}$:\n\n$$\\int -\\frac{2}{7}t^{-5} dt = -\\frac{2}{7} \\times \\frac{t^{-5+1}}{-5+1} = -\\frac{2}{7} \\times \\frac{t^{-4}}{-4}$$\n\nMultiplying the fractions:\n\n$$\\left(-\\frac{2}{7}\\right) \\times \\left(\\frac{t^{-4}}{-4}\\right) = \\frac{2}{28}t^{-4} = \\frac{1}{14}t^{-4} = \\frac{1}{14t^4}$$"},{"type":"text","order":3,"title":"Integrate the constant term","content":"Finally, we integrate the constant term, $\\sqrt{3}$:\n\n$$\\int \\sqrt{3} dt = \\sqrt{3}t$$"},{"type":"text","order":4,"title":"Combine and simplify","content":"Combining the results from our term-by-term integration and remembering to include the constant of integration, $c$, we obtain the general expression for $D(t)$:\n\n$$D(t) = \\frac{1}{8}t^5 + \\frac{1}{14t^4} + \\sqrt{3}t + c$$"}]}],"generatedAt":"2026-06-09T04:46:29.492Z","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":"","difficulty":2,"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\\]","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}} + 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1.9.20 Integrate kx^n

Question 9

1.9.20 Integrate kx^n Questions

1.9.20 Integrate kx^n

Question 9

1.9.20 Integrate kx^n Questions