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

The integral from-3 to 0 of 3x^2e^{x^3} is 1-1/(e^{27)}

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

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

1 - \frac{1}{e ^{27}}

Decimal Notation

1

Solution steps

\int_{- 3}^{0} 3 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

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

Apply u-substitution

: \int_{- 27}^{0} \frac{e ^{u}}{3} d u

= 3 \cdot \int_{- 27}^{0} \frac{e ^{u}}{3} d u

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int_{- 27}^{0} e^{u} d u

Use the common integral:

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

= 3 \cdot \frac{1}{3} [e^{u}]_{- 27}^{0}

Simplify

3 \cdot \frac{1}{3} [e^{u}]_{- 27}^{0} : [e^{u}]_{- 27}^{0}

= [e^{u}]_{- 27}^{0}

Compute the boundaries:

1 - \frac{1}{e ^{27}}

= 1 - \frac{1}{e ^{27}}

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What is the integral from-3 to 0 of 3x^2e^{x^3} ?
The integral from-3 to 0 of 3x^2e^{x^3} is 1-1/(e^{27)}