What is the integral of (2x^2-6)/(x^3+x) ?

The integral of (2x^2-6)/(x^3+x) is 4ln|x^2+1|-6ln|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^2-6)/(x^3+x)

\int \frac{2 x ^{2} - 6}{x ^{3} + x} d x

4 ln x^{2} + 1 - 6 ln ∣ x ∣ + C

Solution steps

\int \frac{2 x ^{2} - 6}{x ^{3} + x} d x

Expand

\frac{2 x ^{2} - 6}{x ^{3} + x} : \frac{2 x ^{2}}{x ^{3} + x} - \frac{6}{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{2 x ^{2}}{x ^{3} + x} d x - \int \frac{6}{x ^{3} + x} d x

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

\int \frac{6}{x ^{3} + x} d x = 6 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

= ln x^{2} + 1 - 6 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1)

Simplify

ln x^{2} + 1 - 6 (ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1) : 4 ln x^{2} + 1 - 6 ln ∣ x ∣

= 4 ln x^{2} + 1 - 6 ln ∣ x ∣

Add a constant to the solution

= 4 ln x^{2} + 1 - 6 ln ∣ x ∣ + C

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

(3, 4), (8, - 3)

x \to 3 lim (2 x^{2} + x - 7)

\int x^{2} - 36 d x

ln (\frac{1 - sin ( x )}{1 + sin ( x )})

What is the integral of (2x^2-6)/(x^3+x) ?
The integral of (2x^2-6)/(x^3+x) is 4ln|x^2+1|-6ln|x|+C