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

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

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

integral

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

2 arctan (2 tan (x)) - x + C

Solution steps

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

Rewrite using trig identities

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

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

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

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

\int 1 d x = x

= 2 arctan (2 tan (x)) - x

Add a constant to the solution

= 2 arctan (2 tan (x)) - x + C

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