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Logarithmic Differentiation

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Question

Solve
$\int \frac{x^{2}}{(x sin x + cos x)^{2}} d x$ .

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Solution

$\int \frac{x^{2} d x}{(x s i n x + c o s x)^{2}}$
$\int \frac{x s i n x d x}{x s i n x + c o s x} - \int \frac{x c o s x (s i n x - x c o s x) d x}{(x s i n x + c o s x)^{2}}$
Using byparts in the second integral,we get-
$u = s i n x - x c o s x$ , $v = \frac{x c o s x}{(x s i n x + c o s x)^{2}}$
$d u = x s i n x$ , $\int v d v = \frac{- 1}{x s i n x + c o s x}$
Second Integral becomes
$- \frac{s i n x - x c o s x}{x s i n x + c o s x} + \int \frac{x s i n x d x}{x s i n x + c o s x}$
Hence the overall equation is,
$\int \frac{x s i n x d x}{x s i n x + c o s x} + \frac{s i n x - x c o s x}{x s i n x + c o s x} - \int \frac{x s i n x d x}{x s i n x + c o s x}$
$⟹ \frac{s i n x - x c o s x}{x s i n x + c o s x}$

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