What is the integral of e^{-x^3}(-3x^2) ?

The integral of e^{-x^3}(-3x^2) is e^{-x^3}+C

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integral of e^{-x^3}(-3x^2)

\int e^{- x^{3}} (- 3 x^{2}) d x

e^{- x^{3}} + C

Solution steps

\int e^{- x^{3}} (- 3 x^{2}) d x

Simplify

= \int - 3 e^{- x^{3}} x^{2} d x

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{e ^{u}}{3} d u

= - 3 \cdot \int - \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 e^{u} d u)

Use the common integral:

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

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

Substitute back

u = - x^{3}

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

Simplify

- 3 (- \frac{1}{3} e^{- x^{3}}) : e^{- x^{3}}

= e^{- x^{3}}

Add a constant to the solution

= e^{- x^{3}} + C

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What is the integral of e^{-x^3}(-3x^2) ?
The integral of e^{-x^3}(-3x^2) is e^{-x^3}+C