What is the integral of (4x)/(sqrt(2x^2+1)) ?

The integral of (4x)/(sqrt(2x^2+1)) is 2sqrt(2x^2+1)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (4x)/(sqrt(2x^2+1))

\int \frac{4 x}{2 x ^{2} + 1} d x

2 2 x^{2} + 1 + C

Solution steps

\int \frac{4 x}{2 x ^{2} + 1} d x

Take the constant out:

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

= 4 \cdot \int \frac{x}{2 x ^{2} + 1} d x

Apply u-substitution

= 4 \cdot \int \frac{1}{4 u} d u

Take the constant out:

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

= 4 \cdot \frac{1}{4} \cdot \int \frac{1}{u} d u

Apply the Power Rule

= 4 \cdot \frac{1}{4} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 2 x^{2} + 1

= 4 \cdot \frac{1}{4} \cdot 2 (2 x^{2} + 1)^{\frac{1}{2}}

Simplify

4 \cdot \frac{1}{4} \cdot 2 (2 x^{2} + 1)^{\frac{1}{2}} : 2 2 x^{2} + 1

= 2 2 x^{2} + 1

Add a constant to the solution

= 2 2 x^{2} + 1 + C

What is the integral of (4x)/(sqrt(2x^2+1)) ?
The integral of (4x)/(sqrt(2x^2+1)) is 2sqrt(2x^2+1)+C