Q.269. Indefinite integration.

Question 269.
Find  ∫x^2 / [x sin (x)+cos(x)]^2 dx.

Answer 269.
x^2 / (xsinx + cosx)^2 dx
= ∫ (x/cosx) * [xcosx / (xsinx + cosx)^2] dx

Using integration by parts,
∫ uv dx = u ∫ vdx - ∫ [(du/dx) ∫ vdx] dx

Taking (x/cosx) = u and (xcosx) / (xsinx + cos)^2 = v
=> integral
= (x/cosx) ∫ (xcosx) / (xsinx + cos)^2 dx - ∫ [d/dx(x/cosx) * ∫ (xcosx) / (xsinx + cos)^2 dx] dx ... (1)

d/dx (x/cosx) = [cosx - x * (-sinx)] / cos^2 x = (xsinx + cos) / cos^2 x ... (2)

∫ (xcosx / (xsinx + cosx)^2 dx
= ∫ (xsinx + cosx)^(-2) * d/dx (xsinx + cosx) dx
= - 1 / (xsinx + cosx) + c ... (3)

Plugging the results (2) and (3) in (1),
Required integral
= (x/cosx) * [- 1/(xsinx + cos)] - ∫ [(xsinx + cos) / cos^2 x] * [- 1/(xsinx + cos)] dx
= - x / [cosx(xsinx + cosx)] + ∫ sec^2 x dx
= - x / [cosx(xsinx + cosx)] + tanx + c
= - x / [cosx(xsinx + cosx)] + sinx/cosx + c
= (- x + xsin^2 x + sinx cosx) / [cosx (xsinx + cosx)] + c
= (-xcos^2 x + sinx cosx) / [cosx (xsinx + cosx)] + c
= (sinx - xcosx) / (cosx + xsinx) + c.

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