What is the integral of (x^5)/(e^{x^6)} ?

The integral of (x^5)/(e^{x^6)} is -1/6 e^{-x^6}+C

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integral of (x^5)/(e^{x^6)}

\int \frac{x ^{5}}{e ^{x^{6}}} d x

- \frac{1}{6} e^{- x^{6}} + C

Solution steps

\int \frac{x ^{5}}{e ^{x^{6}}} d x

Apply exponent rule:

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

\frac{x ^{5}}{e ^{x^{6}}} = (x^{5}) e^{- x^{6}}

= \int x^{5} e^{- x^{6}} d x

Apply u-substitution

= \int - \frac{e ^{u}}{6} d u

Take the constant out:

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

= - \frac{1}{6} \cdot \int e^{u} d u

Use the common integral:

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

= - \frac{1}{6} e^{u}

Substitute back

u = - x^{6}

= - \frac{1}{6} e^{- x^{6}}

Add a constant to the solution

= - \frac{1}{6} e^{- x^{6}} + C

What is the integral of (x^5)/(e^{x^6)} ?
The integral of (x^5)/(e^{x^6)} is -1/6 e^{-x^6}+C