What is the integral of (cos(2x))/(sin^4(x)) ?

The integral of (cos(2x))/(sin^4(x)) is -1/3 cot^3(x)+cot(x)+C

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integral of (cos(2x))/(sin^4(x))

\int \frac{cos ( 2 x )}{sin ^{4} ( x )} d x

- \frac{1}{3} cot^{3} (x) + cot (x) + C

Solution steps

\int \frac{cos ( 2 x )}{sin ^{4} ( x )} d x

Rewrite using trig identities

= \int \frac{cos ^{2} ( x ) - sin ^{2} ( x )}{sin ^{4} ( x )} d x

Expand

\frac{cos ^{2} ( x ) - sin ^{2} ( x )}{sin ^{4} ( x )} : \frac{cos ^{2} ( x )}{sin ^{4} ( x )} - \frac{1}{sin ^{2} ( x )}

= \int \frac{cos ^{2} ( x )}{sin ^{4} ( x )} - \frac{1}{sin ^{2} ( x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{cos ^{2} ( x )}{sin ^{4} ( x )} d x - \int \frac{1}{sin ^{2} ( x )} d x

\int \frac{cos ^{2} ( x )}{sin ^{4} ( x )} d x = - \frac{cot ^{3} ( x )}{3}

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

= - \frac{cot ^{3} ( x )}{3} - (- cot (x))

Simplify

- \frac{cot ^{3} ( x )}{3} - (- cot (x)) : - \frac{1}{3} cot^{3} (x) + cot (x)

= - \frac{1}{3} cot^{3} (x) + cot (x)

Add a constant to the solution

= - \frac{1}{3} cot^{3} (x) + cot (x) + C

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What is the integral of (cos(2x))/(sin^4(x)) ?
The integral of (cos(2x))/(sin^4(x)) is -1/3 cot^3(x)+cot(x)+C