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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

Add a constant to the solution

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

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