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

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

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

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

\frac{1}{1 + sin ( x )} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Apply the Power Rule

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

Substitute back

u = 1 + sin (x)

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

Simplify

= \frac{1}{1 + sin ( x )}

Add a constant to the solution

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

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