What is the integral of (5x^2)*e^{-x^3} ?

The integral of (5x^2)*e^{-x^3} is -5/3 e^{-x^3}+C

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integral of (5x^2)*e^{-x^3}

\int (5 x^{2}) \cdot e^{- x^{3}} d x

- \frac{5}{3} e^{- x^{3}} + C

Solution steps

\int 5 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

= 5 \cdot \int x^{2} e^{- x^{3}} d x

Apply u-substitution

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

= 5 \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

= 5 (- \frac{1}{3} \cdot \int e^{u} d u)

Use the common integral:

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

= 5 (- \frac{1}{3} e^{u})

Substitute back

u = - x^{3}

= 5 (- \frac{1}{3} e^{- x^{3}})

Simplify

5 (- \frac{1}{3} e^{- x^{3}}) : - \frac{5}{3} e^{- x^{3}}

= - \frac{5}{3} e^{- x^{3}}

Add a constant to the solution

= - \frac{5}{3} e^{- x^{3}} + C

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What is the integral of (5x^2)*e^{-x^3} ?
The integral of (5x^2)*e^{-x^3} is -5/3 e^{-x^3}+C