What is the integral of xsqrt(81-x^2) ?

The integral of xsqrt(81-x^2) is -1/3 (81-x^2)^{3/2}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of xsqrt(81-x^2)

\int x 81 - x^{2} d x

- \frac{1}{3} (81 - x^{2})^{\frac{3}{2}} + C

Solution steps

\int x 81 - x^{2} d x

Apply u-substitution

: \int - \frac{u}{2} d u

= \int - \frac{u}{2} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

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

Substitute back

u = 81 - x^{2}

= - \frac{1}{2} \cdot \frac{2}{3} (81 - x^{2})^{\frac{3}{2}}

Simplify

- \frac{1}{2} \cdot \frac{2}{3} (81 - x^{2})^{\frac{3}{2}} : - \frac{1}{3} (81 - x^{2})^{\frac{3}{2}}

= - \frac{1}{3} (81 - x^{2})^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{3} (81 - x^{2})^{\frac{3}{2}} + C

\int_{0}^{12} 2 π x (12 x - x^{2}) d x

(u - u) (u + u)

x^{2} + 2 x - 3

x^{2} y^{^{'}} = 2 x^{2} e^{4 x} - 3 x y

\frac{d}{d t} (\frac{cos ( t ) + sin ( t )}{- sin ( t ) + cos ( t )})

What is the integral of xsqrt(81-x^2) ?
The integral of xsqrt(81-x^2) is -1/3 (81-x^2)^{3/2}+C