What is the integral of x^5e^{1-x^6} ?

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

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integral of x^5e^{1-x^6}

\int x^{5} e^{1 - x^{6}} d x

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

Solution steps

\int x^{5} e^{1 - x^{6}} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{5 e ^{- v + 1}}{6} d v

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

Take the constant out:

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

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

Apply u-substitution

: \int - e^{w} d w

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

Take the constant out:

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

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

Use the common integral:

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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