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

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

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

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

3

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 2 \cdot \int_{- 1}^{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

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

Apply the Power Rule

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

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

Simplify

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

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

Compute the boundaries:

3

= 3

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