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

The integral of x^2(3-x^3)^{2/3} is -1/5 (3-x^3)^{5/3}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

- \frac{1}{5} (3 - x^{3})^{\frac{5}{3}} + C

Solution steps

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

Apply u-substitution

= \int \frac{( 3 - u ) ^{\frac{2}{3}}}{3} d u

Take the constant out:

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

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

Apply u-substitution

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

Take the constant out:

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

= \frac{1}{3} (- \int v^{\frac{2}{3}} d v)

Apply the Power Rule

= \frac{1}{3} (- \frac{3}{5} v^{\frac{5}{3}})

Substitute back

= \frac{1}{3} (- \frac{3}{5} (3 - x^{3})^{\frac{5}{3}})

Simplify

\frac{1}{3} (- \frac{3}{5} (3 - x^{3})^{\frac{5}{3}}) : - \frac{1}{5} (3 - x^{3})^{\frac{5}{3}}

= - \frac{1}{5} (3 - x^{3})^{\frac{5}{3}}

Add a constant to the solution

= - \frac{1}{5} (3 - x^{3})^{\frac{5}{3}} + C

What is the integral of x^2(3-x^3)^{2/3} ?
The integral of x^2(3-x^3)^{2/3} is -1/5 (3-x^3)^{5/3}+C