What is the integral of x^4e^{-6x^5} ?

The integral of x^4e^{-6x^5} is -1/30 e^{-6x^5}+C

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integral of x^4e^{-6x^5}

\int x^{4} e^{- 6 x^{5}} d x

- \frac{1}{30} e^{- 6 x^{5}} + C

Solution steps

\int x^{4} e^{- 6 x^{5}} d x

Apply u-substitution

: \int \frac{e ^{- 6 u^{\frac{5}{4}}} u ^{\frac{1}{4}}}{4} d u

= \int \frac{e ^{- 6 u^{\frac{5}{4}}} u ^{\frac{1}{4}}}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int e^{- 6 u^{\frac{5}{4}}} u^{\frac{1}{4}} d u

Apply u-substitution

: \int \frac{4 e ^{- 6 v}}{5} d v

= \frac{1}{4} \cdot \int \frac{4 e ^{- 6 v}}{5} d v

Take the constant out:

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

= \frac{1}{4} \cdot \frac{4}{5} \cdot \int e^{- 6 v} d v

Apply u-substitution

: \int - \frac{1}{6} e^{w} d w

= \frac{1}{4} \cdot \frac{4}{5} \cdot \int - \frac{1}{6} e^{w} d w

Take the constant out:

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

= \frac{1}{4} \cdot \frac{4}{5} (- \frac{1}{6} \cdot \int e^{w} d w)

Use the common integral:

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

= \frac{1}{4} \cdot \frac{4}{5} (- \frac{1}{6} e^{w})

Substitute back

= \frac{1}{4} \cdot \frac{4}{5} (- \frac{1}{6} e^{- 6 (x^{4})^{\frac{5}{4}}})

Simplify

\frac{1}{4} \cdot \frac{4}{5} (- \frac{1}{6} e^{- 6 (x^{4})^{\frac{5}{4}}}) : - \frac{1}{30} e^{- 6 x^{5}}

= - \frac{1}{30} e^{- 6 x^{5}}

Add a constant to the solution

= - \frac{1}{30} e^{- 6 x^{5}} + C

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What is the integral of x^4e^{-6x^5} ?
The integral of x^4e^{-6x^5} is -1/30 e^{-6x^5}+C