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

The integral of (x^2)/(e^{(x^3))} is -1/3 e^{-x^3}+C

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

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

- \frac{1}{3} e^{- x^{3}} + C

Solution steps

\int \frac{x ^{2}}{e ^{x^{3}}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{x ^{2}}{e ^{x^{3}}} = (x^{2}) e^{- x^{3}}

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

Apply u-substitution

= \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

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

Use the common integral:

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

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

Substitute back

u = - x^{3}

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

Add a constant to the solution

= - \frac{1}{3} e^{- x^{3}} + C

What is the integral of (x^2)/(e^{(x^3))} ?
The integral of (x^2)/(e^{(x^3))} is -1/3 e^{-x^3}+C