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

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

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integral from-1 to 0 of 3x^2e^{-x^3+1}

\int_{- 1}^{0} 3 x^{2} e^{- x^{3} + 1} d x

e^{2} - e

Decimal

4.67077 \dots

Solution steps

\int_{- 1}^{0} 3 x^{2} e^{- x^{3} + 1} d x

Take the constant out:

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

= 3 \cdot \int_{- 1}^{0} x^{2} e^{- x^{3} + 1} d x

Apply u-substitution

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 3 (- \int_{1}^{2} - \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 (- (- \frac{1}{3} \cdot \int_{1}^{2} e^{u} d u))

Use the common integral:

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

= 3 (- (- \frac{1}{3} [e^{u}]_{1}^{2}))

Simplify

3 (- (- \frac{1}{3} [e^{u}]_{1}^{2})) : [e^{u}]_{1}^{2}

= [e^{u}]_{1}^{2}

Compute the boundaries:

e^{2} - e

= e^{2} - e

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