What is the integral of (xcos(x))/(sin^3(x)) ?

The integral of (xcos(x))/(sin^3(x)) is -1/2 xcsc^2(x)-1/2 cot(x)+C

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integral of (xcos(x))/(sin^3(x))

\int \frac{x cos ( x )}{sin ^{3} ( x )} d x

- \frac{1}{2} x csc^{2} (x) - \frac{1}{2} cot (x) + C

Solution steps

\int \frac{x cos ( x )}{sin ^{3} ( x )} d x

Rewrite using trig identities

= \int x cot (x) csc^{2} (x) d x

Apply Integration By Parts

: - \frac{1}{2} x csc^{2} (x) - \int - \frac{1}{2} csc^{2} (x) d x

= - \frac{1}{2} x csc^{2} (x) - \int - \frac{1}{2} csc^{2} (x) d x

\int - \frac{1}{2} csc^{2} (x) d x = \frac{1}{2} cot (x)

= - \frac{1}{2} x csc^{2} (x) - \frac{1}{2} cot (x)

Add a constant to the solution

= - \frac{1}{2} x csc^{2} (x) - \frac{1}{2} cot (x) + C

\int \frac{3}{x ^{2} + 3 x} d x

\int t^{2} sin^{3} (t^{3}) cos^{2} (t^{3}) d t

\int \frac{- x ^{2} - 2}{( x ^{2} + x + 1 ) ^{2}} d x

tan^{2} (x)

\int \frac{1}{( x ^{2} + 1 ) ( x ^{2} + x + 1 )} d x

What is the integral of (xcos(x))/(sin^3(x)) ?
The integral of (xcos(x))/(sin^3(x)) is -1/2 xcsc^2(x)-1/2 cot(x)+C