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

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

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

ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 + 2 arctan (x) + C

Solution steps

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

Take the partial fraction of

\frac{2 x + 1}{x ( x ^{2} + 1 )} : \frac{1}{x} + \frac{- x + 2}{x ^{2} + 1}

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 + 2 arctan (x)

Add a constant to the solution

= ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 + 2 arctan (x) + C

\int x (4 x^{2} - 8) d x

\int s^{3} e^{s} d s

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

\int \frac{1}{1 + cos ( s )} d s

\int \frac{2 x ^{2}}{1 - 8 x ^{3}} d x

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