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

The integral of (x^3+x+1)/(x(x^2+1)) is x+ln|x|-1/2 ln|x^2+1|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^3+x+1)/(x(x^2+1))

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

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

Solution steps

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

Take the partial fraction of

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

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

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

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

Add a constant to the solution

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

y = 9 x, y = x^{2} - 10

x^{4}, at x = - \frac{1}{3}

\int_{0}^{1} 3 e^{- 3 t} d t

\frac{\partial}{\partial x} (lo g_{13} (1 + x^{2}))

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

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