What is the integral of (sin(x)-xcos(x))/(sin^2(x)) ?

The integral of (sin(x)-xcos(x))/(sin^2(x)) is xcot(x)cos(x)+xsin(x)+C

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integral of (sin(x)-xcos(x))/(sin^2(x))

\int \frac{sin ( x ) - x cos ( x )}{sin ^{2} ( x )} d x

x cot (x) cos (x) + x sin (x) + C

Solution steps

\int \frac{sin ( x ) - x cos ( x )}{sin ^{2} ( x )} d x

Apply Integration By Parts

: - cot (x) (sin (x) - x cos (x)) - \int - x sin (x) cot (x) d x

= - cot (x) (sin (x) - x cos (x)) - \int - x sin (x) cot (x) d x

\int - x sin (x) cot (x) d x = - x sin (x) - cos (x)

= - cot (x) (sin (x) - x cos (x)) - (- x sin (x) - cos (x))

Simplify

- cot (x) (sin (x) - x cos (x)) - (- x sin (x) - cos (x)) : x cot (x) cos (x) + x sin (x)

= x cot (x) cos (x) + x sin (x)

Add a constant to the solution

= x cot (x) cos (x) + x sin (x) + C

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What is the integral of (sin(x)-xcos(x))/(sin^2(x)) ?
The integral of (sin(x)-xcos(x))/(sin^2(x)) is xcot(x)cos(x)+xsin(x)+C