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

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

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

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

- cot (x) - x + csc (x) + tan (\frac{x}{2}) + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{cos ( x ) - 1}{cos ( x ) + 1} : \frac{cos ( x )}{cos ( x ) + 1} - \frac{1}{cos ( x ) + 1}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{cos ( x )}{cos ( x ) + 1} d x = cot (x) + x - csc (x)

\int \frac{1}{cos ( x ) + 1} d x = tan (\frac{x}{2})

= - (cot (x) + x - csc (x) - tan (\frac{x}{2}))

Simplify

= - cot (x) - x + csc (x) + tan (\frac{x}{2})

Add a constant to the solution

= - cot (x) - x + csc (x) + tan (\frac{x}{2}) + C

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