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

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

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integral of (2cos(x))/((1+sin(x))^3)

\int \frac{2 cos ( x )}{( 1 + sin ( x ) ) ^{3}} d x

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

Solution steps

\int \frac{2 cos ( x )}{( 1 + sin ( x ) ) ^{3}} d x

Take the constant out:

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

= 2 \cdot \int \frac{cos ( x )}{( 1 + sin ( x ) ) ^{3}} d x

Apply u-substitution

= 2 \cdot \int \frac{1}{u ^{3}} d u

Apply the Power Rule

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

Substitute back

u = 1 + sin (x)

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

Simplify

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

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

Add a constant to the solution

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

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