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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

\frac{2}{3} (x^{3} + 5)^{\frac{3}{2}} + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

u = x^{3} + 5

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

Simplify

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

= \frac{2}{3} (x^{3} + 5)^{\frac{3}{2}}

Add a constant to the solution

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

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