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

The integral of (2x+1)/(1+x^2) is ln|1+x^2|+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 (2x+1)/(1+x^2)

\int \frac{2 x + 1}{1 + x ^{2}} d x

ln 1 + x^{2} + arctan (x) + C

Solution steps

\int \frac{2 x + 1}{1 + x ^{2}} d x

Expand

\frac{2 x + 1}{1 + x ^{2}} : \frac{2 x}{1 + x ^{2}} + \frac{1}{1 + x ^{2}}

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 x}{1 + x ^{2}} d x + \int \frac{1}{1 + x ^{2}} d x

\int \frac{2 x}{1 + x ^{2}} d x = ln 1 + x^{2}

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

= ln 1 + x^{2} + arctan (x)

Add a constant to the solution

= ln 1 + x^{2} + arctan (x) + C

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

\int \frac{5 x + 1}{9 x ^{2} + 6 x + 3} d x

\int x^{3} cos (7 x) d x

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

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