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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

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

= tan (\frac{x}{2}) - (x - sin (x))

Simplify

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

Add a constant to the solution

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

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