What is the integral of (2x^2+3x+5)/((x+1)(x^2+1)) ?

The integral of (2x^2+3x+5)/((x+1)(x^2+1)) is 2ln|x+1|+3arctan(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^2+3x+5)/((x+1)(x^2+1))

\int \frac{2 x ^{2} + 3 x + 5}{( x + 1 ) ( x ^{2} + 1 )} d x

2 ln ∣ x + 1 ∣ + 3 arctan (x) + C

Solution steps

\int \frac{2 x ^{2} + 3 x + 5}{( x + 1 ) ( x ^{2} + 1 )} d x

Take the partial fraction of

\frac{2 x ^{2} + 3 x + 5}{( x + 1 ) ( x ^{2} + 1 )} : \frac{2}{x + 1} + \frac{3}{x ^{2} + 1}

= \int \frac{2}{x + 1} + \frac{3}{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{2}{x + 1} d x + \int \frac{3}{x ^{2} + 1} d x

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

\int \frac{3}{x ^{2} + 1} d x = 3 arctan (x)

= 2 ln ∣ x + 1 ∣ + 3 arctan (x)

Add a constant to the solution

= 2 ln ∣ x + 1 ∣ + 3 arctan (x) + C

\int \frac{2 e ^{x}}{9 - e ^{2 x}} d x

\int \frac{4}{x ^{4} - x ^{2}} d x

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

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

\int sin (x) cos (c) d x

What is the integral of (2x^2+3x+5)/((x+1)(x^2+1)) ?
The integral of (2x^2+3x+5)/((x+1)(x^2+1)) is 2ln|x+1|+3arctan(x)+C