What is the integral of 8x^{13}e^{-x^7} ?

The integral of 8x^{13}e^{-x^7} is 8/7 (-e^{-x^7}x^7-e^{-x^7})+C

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integral of 8x^{13}e^{-x^7}

\int 8 x^{13} e^{- x^{7}} d x

\frac{8}{7} (- e^{- x^{7}} x^{7} - e^{- x^{7}}) + C

Solution steps

\int 8 x^{13} e^{- x^{7}} d x

Take the constant out:

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

= 8 \cdot \int x^{13} e^{- x^{7}} d x

Apply u-substitution

: \int \frac{e ^{- u^{\frac{7}{13}}} u ^{\frac{1}{13}}}{13} d u

= 8 \cdot \int \frac{e ^{- u^{\frac{7}{13}}} u ^{\frac{1}{13}}}{13} d u

Take the constant out:

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

= 8 \cdot \frac{1}{13} \cdot \int e^{- u^{\frac{7}{13}}} u^{\frac{1}{13}} d u

Apply u-substitution

: \int \frac{13 e ^{- v} v}{7} d v

= 8 \cdot \frac{1}{13} \cdot \int \frac{13 e ^{- v} v}{7} d v

Take the constant out:

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

= 8 \cdot \frac{1}{13} \cdot \frac{13}{7} \cdot \int e^{- v} v d v

Apply Integration By Parts

: - e^{- v} v - \int - e^{- v} d v

= 8 \cdot \frac{1}{13} \cdot \frac{13}{7} (- e^{- v} v - \int - e^{- v} d v)

\int - e^{- v} d v = e^{- v}

= 8 \cdot \frac{1}{13} \cdot \frac{13}{7} (- e^{- v} v - e^{- v})

Substitute back

= 8 \cdot \frac{1}{13} \cdot \frac{13}{7} (- e^{- (x^{13})^{\frac{7}{13}}} (x^{13})^{\frac{7}{13}} - e^{- (x^{13})^{\frac{7}{13}}})

Simplify

8 \cdot \frac{1}{13} \cdot \frac{13}{7} (- e^{- (x^{13})^{\frac{7}{13}}} (x^{13})^{\frac{7}{13}} - e^{- (x^{13})^{\frac{7}{13}}}) : \frac{8}{7} (- e^{- x^{7}} x^{7} - e^{- x^{7}})

= \frac{8}{7} (- e^{- x^{7}} x^{7} - e^{- x^{7}})

Add a constant to the solution

= \frac{8}{7} (- e^{- x^{7}} x^{7} - e^{- x^{7}}) + C

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What is the integral of 8x^{13}e^{-x^7} ?
The integral of 8x^{13}e^{-x^7} is 8/7 (-e^{-x^7}x^7-e^{-x^7})+C