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

The integral of (5x)/(5x^2+2) is 1/2 ln|5x^2+2|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{1}{2} ln 5 x^{2} + 2 + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{10 u} d u

= 5 \cdot \int \frac{1}{10 u} d u

Take the constant out:

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

= 5 \cdot \frac{1}{10} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 \cdot \frac{1}{10} ln ∣ u ∣

Substitute back

u = 5 x^{2} + 2

= 5 \cdot \frac{1}{10} ln 5 x^{2} + 2

Simplify

5 \cdot \frac{1}{10} ln 5 x^{2} + 2 : \frac{1}{2} ln 5 x^{2} + 2

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

Add a constant to the solution

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

f (x) = e^{6} + ln (4)

\frac{d}{d x} (4 x^{3} - 24 x^{2} + 36 x)

\frac{d P}{d t} = 1 P (500 - P)

- sin (3 x)

x \cdot y^{^{'}} + y = \frac{1}{y ^{2}}

What is the integral of (5x)/(5x^2+2) ?
The integral of (5x)/(5x^2+2) is 1/2 ln|5x^2+2|+C