What is the integral from 0 to 1 of x^3sqrt(1-x^2) ?

The integral from 0 to 1 of x^3sqrt(1-x^2) is 2/15

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from 0 to 1 of x^3sqrt(1-x^2)

\int_{0}^{1} x^{3} 1 - x^{2} d x

\frac{2}{15}

Decimal

0.13333 \dots

Solution steps

\int_{0}^{1} x^{3} 1 - x^{2} d x

Apply u-substitution

: \int_{1}^{0} - \frac{( - u + 1 ) u}{2} d u

= \int_{1}^{0} - \frac{( - u + 1 ) u}{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{0}^{1} - \frac{( - u + 1 ) 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_{0}^{1} (- u + 1) u d u)

Expand

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

= - (- \frac{1}{2} \cdot \int_{0}^{1} - u^{\frac{3}{2}} + u d u)

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- \frac{1}{2} (- \int_{0}^{1} u^{\frac{3}{2}} d u + \int_{0}^{1} u d u))

\int_{0}^{1} u^{\frac{3}{2}} d u = \frac{2}{5}

\int_{0}^{1} u d u = \frac{2}{3}

= - (- \frac{1}{2} (- \frac{2}{5} + \frac{2}{3}))

Simplify

- (- \frac{1}{2} (- \frac{2}{5} + \frac{2}{3})) : \frac{2}{15}

= \frac{2}{15}

- \frac{2}{3 x ^{2}}

\frac{\partial}{\partial x} (2 x y - 3 y^{2})

\frac{d}{d x} ((x^{3} + 3) 2^{- 7 x})

\frac{d}{d x} (24 x^{2} + 8 x - 5)

\frac{d}{d t} (2 t + 3)

What is the integral from 0 to 1 of x^3sqrt(1-x^2) ?
The integral from 0 to 1 of x^3sqrt(1-x^2) is 2/15