What is the integral of (cos(2x))/((2+3sin(2x))^3) ?

The integral of (cos(2x))/((2+3sin(2x))^3) is -1/(12(2+3sin(2x))^2)+C

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integral of (cos(2x))/((2+3sin(2x))^3)

\int \frac{cos ( 2 x )}{( 2 + 3 sin ( 2 x ) ) ^{3}} d x

- \frac{1}{12 ( 2 + 3 sin ( 2 x ) ) ^{2}} + C

Solution steps

\int \frac{cos ( 2 x )}{( 2 + 3 sin ( 2 x ) ) ^{3}} d x

Apply u-substitution

: \int \frac{1}{6 u ^{3}} d u

= \int \frac{1}{6 u ^{3}} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{u ^{3}} d u

Apply the Power Rule

: - \frac{1}{2 u ^{2}}

= \frac{1}{6} (- \frac{1}{2 u ^{2}})

Substitute back

u = 2 + 3 sin (2 x)

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

Simplify

\frac{1}{6} (- \frac{1}{2 ( 2 + 3 sin ( 2 x ) ) ^{2}}) : - \frac{1}{12 ( 2 + 3 sin ( 2 x ) ) ^{2}}

= - \frac{1}{12 ( 2 + 3 sin ( 2 x ) ) ^{2}}

Add a constant to the solution

= - \frac{1}{12 ( 2 + 3 sin ( 2 x ) ) ^{2}} + C

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What is the integral of (cos(2x))/((2+3sin(2x))^3) ?
The integral of (cos(2x))/((2+3sin(2x))^3) is -1/(12(2+3sin(2x))^2)+C