What is the integral of (cos(2x))/((1+sin(2x))^4) ?

The integral of (cos(2x))/((1+sin(2x))^4) is -1/(6(1+sin(2x))^3)+C

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integral of (cos(2x))/((1+sin(2x))^4)

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

- \frac{1}{6 ( 1 + sin ( 2 x ) ) ^{3}} + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 u ^{4}} d u

= \int \frac{1}{2 u ^{4}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{1}{u ^{4}} d u

Apply the Power Rule

: - \frac{1}{3 u ^{3}}

= \frac{1}{2} (- \frac{1}{3 u ^{3}})

Substitute back

u = 1 + sin (2 x)

= \frac{1}{2} (- \frac{1}{3 ( 1 + sin ( 2 x ) ) ^{3}})

Simplify

\frac{1}{2} (- \frac{1}{3 ( 1 + sin ( 2 x ) ) ^{3}}) : - \frac{1}{6 ( 1 + sin ( 2 x ) ) ^{3}}

= - \frac{1}{6 ( 1 + sin ( 2 x ) ) ^{3}}

Add a constant to the solution

= - \frac{1}{6 ( 1 + sin ( 2 x ) ) ^{3}} + C

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What is the integral of (cos(2x))/((1+sin(2x))^4) ?
The integral of (cos(2x))/((1+sin(2x))^4) is -1/(6(1+sin(2x))^3)+C