What is the integral of (cos(x))/((2-3sin(x))^4) ?

The integral of (cos(x))/((2-3sin(x))^4) is 1/(9(2-3sin(x))^3)+C

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integral of (cos(x))/((2-3sin(x))^4)

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

\frac{1}{9 ( 2 - 3 sin ( x ) ) ^{3}} + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{( 2 - 3 u ) ^{4}} d u

= \int \frac{1}{( 2 - 3 u ) ^{4}} d u

Apply u-substitution

: \int - \frac{1}{3 v ^{4}} d v

= \int - \frac{1}{3 v ^{4}} d v

Take the constant out:

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

= - \frac{1}{3} \cdot \int \frac{1}{v ^{4}} d v

Apply the Power Rule

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

= - \frac{1}{3} (- \frac{1}{3 v ^{3}})

Substitute back

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

Simplify

- \frac{1}{3} (- \frac{1}{3 ( 2 - 3 sin ( x ) ) ^{3}}) : \frac{1}{9 ( 2 - 3 sin ( x ) ) ^{3}}

= \frac{1}{9 ( 2 - 3 sin ( x ) ) ^{3}}

Add a constant to the solution

= \frac{1}{9 ( 2 - 3 sin ( x ) ) ^{3}} + C

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What is the integral of (cos(x))/((2-3sin(x))^4) ?
The integral of (cos(x))/((2-3sin(x))^4) is 1/(9(2-3sin(x))^3)+C