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

The integral of 2(2+x^2)^{3/2}x is 2/5 (2+x^2)^{5/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of 2(2+x^2)^{3/2}x

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

\frac{2}{5} (2 + x^{2})^{\frac{5}{2}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

= 2 \cdot \frac{1}{2} \cdot \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = 2 + x^{2}

= 2 \cdot \frac{1}{2} \cdot \frac{2}{5} (2 + x^{2})^{\frac{5}{2}}

Simplify

2 \cdot \frac{1}{2} \cdot \frac{2}{5} (2 + x^{2})^{\frac{5}{2}} : \frac{2}{5} (2 + x^{2})^{\frac{5}{2}}

= \frac{2}{5} (2 + x^{2})^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{5} (2 + x^{2})^{\frac{5}{2}} + C

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