What is the integral from 0 to 1 of 3t^2e^{t^3} ?

The integral from 0 to 1 of 3t^2e^{t^3} is e-1

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integral from 0 to 1 of 3t^2e^{t^3}

\int_{0}^{1} 3 t^{2} e^{t^{3}} d t

e - 1

Decimal

1.71828 \dots

Solution steps

\int_{0}^{1} 3 t^{2} e^{t^{3}} d t

Take the constant out:

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

= 3 \cdot \int_{0}^{1} t^{2} e^{t^{3}} d t

Apply u-substitution

: \int_{0}^{1} \frac{e ^{u}}{3} d u

= 3 \cdot \int_{0}^{1} \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_{0}^{1} e^{u} d u

Use the common integral:

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

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

Simplify

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

= [e^{u}]_{0}^{1}

Compute the boundaries:

e - 1

= e - 1

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What is the integral from 0 to 1 of 3t^2e^{t^3} ?
The integral from 0 to 1 of 3t^2e^{t^3} is e-1