What is the integral of (3x)/(x^2+5) ?

The integral of (3x)/(x^2+5) is 3/2 ln|x^2+5|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (3x)/(x^2+5)

\int \frac{3 x}{x ^{2} + 5} d x

\frac{3}{2} ln x^{2} + 5 + C

Solution steps

\int \frac{3 x}{x ^{2} + 5} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{x}{x ^{2} + 5} d x

Apply u-substitution

= 3 \cdot \int \frac{1}{2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = x^{2} + 5

= 3 \cdot \frac{1}{2} ln x^{2} + 5

Simplify

3 \cdot \frac{1}{2} ln x^{2} + 5 : \frac{3}{2} ln x^{2} + 5

= \frac{3}{2} ln x^{2} + 5

Add a constant to the solution

= \frac{3}{2} ln x^{2} + 5 + C

What is the integral of (3x)/(x^2+5) ?
The integral of (3x)/(x^2+5) is 3/2 ln|x^2+5|+C