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

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

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integral of (4cos(x))/((3+sin(x))^3)

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

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

Solution steps

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

Take the constant out:

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

= 4 \cdot \int \frac{cos ( x )}{( 3 + sin ( x ) ) ^{3}} d x

Apply u-substitution

: \int \frac{1}{( 3 + u ) ^{3}} d u

= 4 \cdot \int \frac{1}{( 3 + u ) ^{3}} d u

Apply u-substitution

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

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

Apply the Power Rule

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

= 4 (- \frac{1}{2 v ^{2}})

Substitute back

= 4 (- \frac{1}{2 ( 3 + sin ( x ) ) ^{2}})

Simplify

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

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

Add a constant to the solution

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

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