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

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

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

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

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

Solution steps

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

Simplify

= \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

\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

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