What is the integral from 0 to 2 of 3x(x^2+2)^3 ?

The integral from 0 to 2 of 3x(x^2+2)^3 is 480

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integral from 0 to 2 of 3x(x^2+2)^3

\int_{0}^{2} 3 x (x^{2} + 2)^{3} d x

480

Solution steps

\int_{0}^{2} 3 x (x^{2} + 2)^{3} d x

Take the constant out:

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

= 3 \cdot \int_{0}^{2} x (x^{2} + 2)^{3} d x

Apply u-substitution

: \int_{2}^{6} \frac{u ^{3}}{2} d u

= 3 \cdot \int_{2}^{6} \frac{u ^{3}}{2} d u

Take the constant out:

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

= 3 \cdot \frac{1}{2} \cdot \int_{2}^{6} u^{3} d u

Apply the Power Rule

: [\frac{u ^{4}}{4}]_{2}^{6}

= 3 \cdot \frac{1}{2} [\frac{u ^{4}}{4}]_{2}^{6}

Simplify

3 \cdot \frac{1}{2} [\frac{u ^{4}}{4}]_{2}^{6} : \frac{3}{2} [\frac{u ^{4}}{4}]_{2}^{6}

= \frac{3}{2} [\frac{u ^{4}}{4}]_{2}^{6}

Compute the boundaries:

320

= \frac{3}{2} \cdot 320

Simplify

= 480

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What is the integral from 0 to 2 of 3x(x^2+2)^3 ?
The integral from 0 to 2 of 3x(x^2+2)^3 is 480