What is the integral of (sec(x))/(2tan^2(x)+1) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (sec(x))/(2tan^2(x)+1)

\int \frac{sec ( x )}{2 tan ^{2} ( x ) + 1} d x

arctan (sin (x)) + C

Solution steps

\int \frac{sec ( x )}{2 tan ^{2} ( x ) + 1} d x

Rewrite using trig identities

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

Rewrite using trig identities

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

Apply u-substitution

= \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= arctan (u)

Substitute back

u = sin (x)

= arctan (sin (x))

Add a constant to the solution

= arctan (sin (x)) + C

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