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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

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

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

\int 1 d x = x

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

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

Add a constant to the solution

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

\int 15 x^{3} e^{x} d x

x \to \infty lim (- x)

\frac{d}{d x} (8 x arcsin (x))

\frac{d}{d x} (3 x ln (9 x^{2}))

x \to 0 lim (e^{(\frac{1}{x})})

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