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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Expand

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int \frac{36 x ^{2} - 25}{x ^{3}} d x

\int \frac{sin ( x )}{cos ^{3} ( x ) - cos ( x )} d x

\int \frac{3 + x}{x ^{2023}} d x

x + 1

\int (\frac{1}{2} x^{4} + \frac{1}{4} x^{3} - x) d x

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