A calculus problem by Superman

Calculus Level 3

∫ x 2 ( x cos ⁡ x − sin ⁡ x ) ( x sin ⁡ x + cos ⁡ x ) d x = ? \int \dfrac{x^2}{(x \cos x-\sin x)(x \sin x+\cos x)} \, dx = \, ?

Clarification : C C denotes the arbitrary constant of integration .

ln ⁡ ∣ x sin ⁡ x − cos ⁡ x ∣ ∣ x sin ⁡ x + cos ⁡ x ∣ + C \ln\dfrac{\mid x\sin x-\cos x \mid}{\mid x\sin x+\cos x \mid}+C ln ⁡ ∣ x cos ⁡ x + sin ⁡ x ∣ ∣ x sin ⁡ x − cos ⁡ x ∣ + C \ln\dfrac{\mid x\cos x+\sin x \mid}{\mid x\sin x-\cos x \mid}+C ln ⁡ ∣ x sin ⁡ x + cos ⁡ x ∣ ∣ x cos ⁡ x − sin ⁡ x ∣ + C \ln\dfrac{\mid x\sin x+\cos x \mid}{\mid x\cos x-\sin x \mid}+C ln ⁡ ∣ x sin ⁡ x − cos ⁡ x ∣ ∣ x cos ⁡ x − sin ⁡ x ∣ + C \ln\dfrac{\mid x\sin x-\cos x \mid}{\mid x\cos x-\sin x \mid}+C

Rudraksh Shukla by Rudraksh Shukla , 23, India

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3 solutions

Sam Andrews
Feb 17, 2019

Multiply numerator by [Sinx(xsinx + cosx) + cosx(xcosx-sinx) ]= x. Hence multiply denominator by x. Now decompose it into two fraction and you get the answer in the very next step.

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Rakshith Lokesh
Apr 1, 2018

hint : the expressions in denomino are derivatives of each other

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Try decomposing the given fraction into two terms of the form A(x)/(xcosx-sinx) and B(x)/(xsinx+cosx). Summing them and multuplying out will make clear that A(x)=xsinx and B(x)=xcosx works. Now it's simple.

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