t) = 4\\sin^3(2\\theta) \\cdot \\frac{d}{d\\theta}(\\sin 2\\theta) = 4\\sin^3(2\\theta) \\cdot 2\\cos(2\\theta) = 8\\sin^3(2\\theta)\\cos(2\\theta)$$\n\nWe need our derivative to equal $12\\sin^3(2\\theta)\\cos(2\\theta)$. To adjust for the constant, we multiply by $\\frac{12}{8} = \\frac{3}{2}$:\n\n$$\\frac{d}{d\\theta} \\left( \\frac{3}{2}\\sin^4(2\\theta) \\right) = \\frac{3}{2} \\cdot 8\\sin^3(2\\theta)\\cos(2\\theta) = 12\\sin^3(2\\theta)\\cos(2\\theta)$$\n\nThus, by the reverse chain rule, the antiderivative is:\n\n$$\\frac{3}{2}\\sin^4(2\\theta) + C$$"}]}],"generatedAt":"2026-06-11T03:45:34.049Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"dd3bc52d-8e70-450b-9216-5b01549665bd","revisionId":"6ca74116-a192-4d79-b9eb-dffc544e277e","questionNumber":null,"questionType":"LA","specification":"","difficulty":5,"difficultyBand":"medium","totalMarks":5,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"60e32332-9c7a-4f5f-a463-5d6f6e35adc0","label":null,"content":"Find, in simplest form,\n\n\\[\\int (3\\cos x + \\sin x)^2 \\, dx\\]","markscheme":"- Expands the integrand to $9\\cos^2 x + 6\\sin x \\cos x + \\sin^2 x$ M1\n- Uses the identity $\\sin 2x = 2\\sin x \\cos x$ to rewrite the term $6\\sin x \\cos x$ as $3\\sin 2x$ M1\n- Uses the double angle identity $\\cos^2 x = \\frac{1 + \\cos 2x}{2}$ (and potentially $\\sin^2 x = \\frac{1 - \\cos 2x}{2}$) to rewrite squared terms in terms of $\\cos 2x$ M1\n- Obtains the simplified integrand $5 + 4\\cos 2x + 3\\sin 2x$ A1\n- Integrates to obtain the final result $5x + 2\\sin 2x - \\frac{3}{2}\\cos 2x + C$ A1","marks":5,"order":0,"markingRubric":null,"explanation":{"methods":[{"id":"double_angle_expansion","label":"Trigonometric Expansion and Double-Angle Identities","steps":[{"type":"text","order":0,"title":"Expand the integrand","content":"To solve this integral using trigonometric identities, we first need to expand the squared expression inside the integral. \n\nRecall the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$. \n\nLetting $a = 3\\cos x$ and $b = \\sin x$, we expand the integrand:\n\n$$(3\\cos x + \\sin x)^2 = 9\\cos^2 x + 6\\sin x \\cos x + \\sin^2 x$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\sin 2x = 2\\sin x \\cos x \\implies \\htmlData{anchor=middle-term}{\\color{@sky}{6\\sin x \\cos x}} = \\htmlData{anchor=middle-term-sub}{\\color{@sky}{3\\sin 2x}}$"},{"latex":"$$\\cos^2 x = \\htmlData{anchor=cos-sq}{\\color{@amber}{\\frac{1 + \\cos 2x}{2}}}, \\quad \\sin^2 x = \\htmlData{anchor=sin-sq}{\\color{@purple}{\\frac{1 - \\cos 2x}{2}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"middle-term"},{"color":"amber","style":"box","anchor":"cos-sq"},{"color":"purple","style":"box","anchor":"sin-sq"}],"connections":[{"to":"middle-term-sub","from":"middle-term","color":"sky","label":"double-angle","style":"solid"}]},"order":1,"title":"Apply double-angle formulas","description":"Next, we use standard double-angle identities from the A-Level formula booklet to linearise the squared and product trigonometric terms."},{"type":"text","order":2,"title":"Simplify the integrand","content":"Now we substitute these identities into our expanded expression to simplify the integrand:\n\n$$\\begin{aligned} 9\\cos^2 x + 6\\sin x \\cos x + \\sin^2 x &= 9\\left(\\frac{1 + \\cos 2x}{2}\\right) + 3\\sin 2x + \\left(\\frac{1 - \\cos 2x}{2}\\right) \\\\ &= \\frac{9 + 9\\cos 2x}{2} + \\frac{1 - \\cos 2x}{2} + 3\\sin 2x \\\\ &= \\frac{10 + 8\\cos 2x}{2} + 3\\sin 2x \\\\ &= 5 + 4\\cos 2x + 3\\sin 2x \\end{aligned}$$\n\nThis simplified form is much easier to integrate because all trigonometric terms are now linear."},{"type":"text","order":3,"title":"Integrate term by term","content":"Now we can rewrite the original integral and integrate each term individually using standard integration rules:\n\n$$\\int (3\\cos x + \\sin x)^2 \\, dx = \\int (5 + 4\\cos 2x + 3\\sin 2x) \\, dx$$\n\nIntegrating each term with respect to $x$:\n- The integr

1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only)

Question 32

1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only) Questions

1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only)

Question 32

1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only) Questions