What is the integral from-1 to 0 of (x(x^2-1)^{10}) ?

The integral from-1 to 0 of (x(x^2-1)^{10}) is -1/22

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integral from-1 to 0 of (x(x^2-1)^{10})

\int_{- 1}^{0} (x (x^{2} - 1)^{10}) d x

- \frac{1}{22}

Decimal Notation

- 0.04545 \dots

Solution steps

\int_{- 1}^{0} x (x^{2} - 1)^{10} d x

Apply u-substitution

: \int_{0}^{- 1} \frac{u ^{10}}{2} d u

= \int_{0}^{- 1} \frac{u ^{10}}{2} d u

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

= - \int_{- 1}^{0} \frac{u ^{10}}{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_{- 1}^{0} u^{10} d u

Apply the Power Rule

: [\frac{u ^{11}}{11}]_{- 1}^{0}

= - \frac{1}{2} [\frac{u ^{11}}{11}]_{- 1}^{0}

Compute the boundaries:

\frac{1}{11}

= - \frac{1}{2} \cdot \frac{1}{11}

Simplify

= - \frac{1}{22}

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What is the integral from-1 to 0 of (x(x^2-1)^{10}) ?
The integral from-1 to 0 of (x(x^2-1)^{10}) is -1/22