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

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

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

\frac{3}{2} ln x^{2} + 1 - \frac{x ^{3}}{x ^{2} + 1} - \frac{x}{x ^{2} + 1} - ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

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

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

= - (\frac{x ^{3}}{x ^{2} + 1} - arctan (x) + \frac{x}{1 + x ^{2}}) + ln x^{2} + 1 - arctan (x) - (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

Simplify

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

= \frac{3}{2} ln x^{2} + 1 - \frac{x ^{3}}{x ^{2} + 1} - \frac{x}{x ^{2} + 1} - ln ∣ x ∣

Add a constant to the solution

= \frac{3}{2} ln x^{2} + 1 - \frac{x ^{3}}{x ^{2} + 1} - \frac{x}{x ^{2} + 1} - ln ∣ x ∣ + C

x \to 0 + lim (\frac{x \cdot ln ( x ) + 2}{x})

x \to 0 - lim (x e^{\frac{2}{x}} + 1)

\int_{- 1}^{3} 3 x^{2} d x

f (x) = I n (x)

x \to \infty lim (3 - \frac{1}{x})

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