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

Add a constant to the solution

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

\frac{d}{d x} (\frac{4 x}{4 + x})

\int \frac{1}{e ^{x} + 2} d x

\frac{d}{d x} (e^{- x^{2}} - sin (x) cos (x))

\frac{\partial}{\partial x} (ln (2 x^{3} + y))

f (x) = (- 25 x^{2} + 9)^{2}

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