What is the integral of (2x+5)/(x^2+1) ?

The integral of (2x+5)/(x^2+1) is ln|x^2+1|+5arctan(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (2x+5)/(x^2+1)

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

ln x^{2} + 1 + 5 arctan (x) + C

Solution steps

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

Expand

\frac{2 x + 5}{x ^{2} + 1} : \frac{2 x}{x ^{2} + 1} + \frac{5}{x ^{2} + 1}

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

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

\int \frac{5}{x ^{2} + 1} d x = 5 arctan (x)

= ln x^{2} + 1 + 5 arctan (x)

Add a constant to the solution

= ln x^{2} + 1 + 5 arctan (x) + C

What is the integral of (2x+5)/(x^2+1) ?
The integral of (2x+5)/(x^2+1) is ln|x^2+1|+5arctan(x)+C