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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/((1+x^2)(1+arctan^2(x)))

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

arctan (arctan (x)) + C

Solution steps

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

Apply u-substitution

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

= \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 = arctan (x)

= arctan (arctan (x))

Add a constant to the solution

= arctan (arctan (x)) + C

\int \frac{5}{e ^{2 x}} d x

\int \frac{1}{16 - 6 x - x ^{2}} d x

\int x^{- \frac{7}{8}} d x

\int \frac{1}{( y ^{2} + 7 ) ^{\frac{3}{2}}} d y

\int 2 \cdot \frac{ln ( x )}{x ^{4}} d x

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