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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\int \frac{3 x + 2}{x ^{2} + 9} d x

\int \frac{y ^{2} - 1}{y} d y

\int - \frac{5}{x ^{2} 5 + x ^{2}} d x

\int 8 x sin (5 - 4 x) d x

\int 3 (x + 4)^{2} d x

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