What is the integral from 0 to 2 of 4x^2e^{x^3} ?

The integral from 0 to 2 of 4x^2e^{x^3} is (4(e^8-1))/3

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integral from 0 to 2 of 4x^2e^{x^3}

\int_{0}^{2} 4 x^{2} e^{x^{3}} d x

\frac{4 ( e ^{8} - 1 )}{3}

Decimal Notation

3973.27731 \dots

Solution steps

\int_{0}^{2} 4 x^{2} e^{x^{3}} d x

Take the constant out:

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

= 4 \cdot \int_{0}^{2} x^{2} e^{x^{3}} d x

Apply u-substitution

: \int_{0}^{8} \frac{e ^{u}}{3} d u

= 4 \cdot \int_{0}^{8} \frac{e ^{u}}{3} d u

Take the constant out:

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

= 4 \cdot \frac{1}{3} \cdot \int_{0}^{8} e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= 4 \cdot \frac{1}{3} [e^{u}]_{0}^{8}

Simplify

4 \cdot \frac{1}{3} [e^{u}]_{0}^{8} : \frac{4}{3} [e^{u}]_{0}^{8}

= \frac{4}{3} [e^{u}]_{0}^{8}

Compute the boundaries:

e^{8} - 1

= \frac{4}{3} (e^{8} - 1)

Simplify

= \frac{4 ( e ^{8} - 1 )}{3}

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What is the integral from 0 to 2 of 4x^2e^{x^3} ?
The integral from 0 to 2 of 4x^2e^{x^3} is (4(e^8-1))/3