What is the integral of (xsqrt(1-x^2)) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (xsqrt(1-x^2))

\int (x 1 - x^{2}) d x

- \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} + C

Solution steps

\int x 1 - x^{2} d x

Apply u-substitution

= \int - u^{2} d u

Take the constant out:

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

= - \int u^{2} d u

Apply the Power Rule

= - \frac{u ^{3}}{3}

Substitute back

u = 1 - x^{2}

= - \frac{( 1 - x ^{2} ) ^{3}}{3}

Simplify

- \frac{( 1 - x ^{2} ) ^{3}}{3} : - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}}

= - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} + C

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