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

The integral from 0 to 2 of x(x^2-2)^2 is 8/3

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

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

\frac{8}{3}

Decimal

2.66666 \dots

Solution steps

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

Apply u-substitution

: \int_{- 2}^{2} \frac{u ^{2}}{2} d u

= \int_{- 2}^{2} \frac{u ^{2}}{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_{- 2}^{2} u^{2} d u

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{- 2}^{2}

= \frac{1}{2} [\frac{u ^{3}}{3}]_{- 2}^{2}

Compute the boundaries:

\frac{16}{3}

= \frac{1}{2} \cdot \frac{16}{3}

Simplify

= \frac{8}{3}

\int_{- 1}^{1} \frac{x ^{2}}{1 + x ^{2}} d x

\int_{- 6}^{6} x^{2} - 25 d x

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\int_{1}^{7} x^{2} - 2 x - 15 d x

\int_{0}^{7} x^{2} d x

What is the integral from 0 to 2 of x(x^2-2)^2 ?
The integral from 0 to 2 of x(x^2-2)^2 is 8/3