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

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

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

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

2 x + \frac{2}{tan ( \frac{x}{2} ) + 1} + cos (x) + C

Solution steps

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

Expand

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

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

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

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

Simplify

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

Add a constant to the solution

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

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