What is the integral from 2 to 4 of 2x(4+x^2)^3 ?

The integral from 2 to 4 of 2x(4+x^2)^3 is 38976

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integral from 2 to 4 of 2x(4+x^2)^3

\int_{2}^{4} 2 x (4 + x^{2})^{3} d x

38976

Solution steps

\int_{2}^{4} 2 x (4 + x^{2})^{3} d x

Take the constant out:

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

= 2 \cdot \int_{2}^{4} x (4 + x^{2})^{3} d x

Apply u-substitution

: \int_{8}^{20} \frac{u ^{3}}{2} d u

= 2 \cdot \int_{8}^{20} \frac{u ^{3}}{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_{8}^{20} u^{3} d u

Apply the Power Rule

: [\frac{u ^{4}}{4}]_{8}^{20}

= 2 \cdot \frac{1}{2} [\frac{u ^{4}}{4}]_{8}^{20}

Simplify

2 \cdot \frac{1}{2} [\frac{u ^{4}}{4}]_{8}^{20} : [\frac{u ^{4}}{4}]_{8}^{20}

= [\frac{u ^{4}}{4}]_{8}^{20}

Compute the boundaries:

38976

= 38976

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What is the integral from 2 to 4 of 2x(4+x^2)^3 ?
The integral from 2 to 4 of 2x(4+x^2)^3 is 38976