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Exercise 11.5 · Q3

Q.(2x−5)(36+4x)(2x-5)(36+4x)

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Concept understanding — Integration by Decomposition and Partial Fractions

Some integrands have no direct standard formula but can be decomposed into a sum or difference of functions whose integrals are known, then integrated term by term. Typical decompositions: expanding a power such as (3x−1)2(3x-1)^2, splitting a single fraction like x2−3x+1x\dfrac{x^2-3x+1}{x} into separate terms, and using trigonometric product-to-sum identities (e.g. 2cos⁡Asin⁡B=sin⁡(A+B)−sin⁡(A−B)2\cos A\sin B=\sin(A+B)-\sin(A-B), cos⁡3x=14(3cos⁡x+cos⁡3x)\cos^3 x=\tfrac14(3\cos x+\cos 3x)), power-reduction (1+cos⁡2x=2cos⁡2x1+\cos 2x=2\cos^2 x, 1−cos⁡2x=2sin⁡2x1-\cos 2x=2\sin^2 x), and sin⁡2x+cos⁡2x=1\sin^2 x+\cos^2 x=1.

Partial fractions. When the integrand is a rational function p(x)q(x)\dfrac{p(x)}{q(x)} with deg⁡p<deg⁡q\deg p<\deg q (if not, first do polynomial long division), resolve it into a sum of simpler fractions — one term per distinct linear factor Ax−α\dfrac{A}{x-\alpha} (and B(x−α)2\dfrac{B}{(x-\alpha)^2} for a repeated factor) — determine the constants by comparing coefficients or substituting convenient xx-values, then integrate each simple fraction (each gives a logarithm or a power).

Tip

If the numerator's degree is ≥\ge the denominator's, you must divide first; only the proper-fraction remainder is expanded into partial fractions.

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