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

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

Popular >

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

\int_{0}^{1} x (1 - x^{2}) d x

\frac{1}{3}

Decimal Notation

0.33333 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{0} - u^{2} d u

= \int_{1}^{0} - u^{2} d u

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

= - \int_{0}^{1} - u^{2} d u

Take the constant out:

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

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

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{0}^{1}

= - (- [\frac{u ^{3}}{3}]_{0}^{1})

Simplify

= [\frac{u ^{3}}{3}]_{0}^{1}

Compute the boundaries:

\frac{1}{3}

= \frac{1}{3}

\int_{1}^{\infty} n^{- 3}

\int_{0}^{1} x (x - 1)^{4} d x

\int_{0}^{π} 1 sin (x) d x

\int_{- \frac{π}{2}}^{\frac{π}{2}} 5 cos (t) d t

\int_{0}^{x} \frac{3}{( 1 + x ) ^{4}} d x

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