What is the integral of 3x^{15}e^{-x^8} ?

The integral of 3x^{15}e^{-x^8} is 3/8 (-e^{-x^8}x^8-e^{-x^8})+C

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integral of 3x^{15}e^{-x^8}

\int 3 x^{15} e^{- x^{8}} d x

\frac{3}{8} (- e^{- x^{8}} x^{8} - e^{- x^{8}}) + C

Solution steps

\int 3 x^{15} e^{- x^{8}} d x

Take the constant out:

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

= 3 \cdot \int x^{15} e^{- x^{8}} d x

Apply u-substitution

: \int \frac{e ^{u} u}{8} d u

= 3 \cdot \int \frac{e ^{u} u}{8} d u

Take the constant out:

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

= 3 \cdot \frac{1}{8} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= 3 \cdot \frac{1}{8} (e^{u} u - \int e^{u} d u)

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

= 3 \cdot \frac{1}{8} (e^{u} u - e^{u})

Substitute back

u = - x^{8}

= 3 \cdot \frac{1}{8} (e^{- x^{8}} (- x^{8}) - e^{- x^{8}})

Simplify

3 \cdot \frac{1}{8} (e^{- x^{8}} (- x^{8}) - e^{- x^{8}}) : \frac{3}{8} (- e^{- x^{8}} x^{8} - e^{- x^{8}})

= \frac{3}{8} (- e^{- x^{8}} x^{8} - e^{- x^{8}})

Add a constant to the solution

= \frac{3}{8} (- e^{- x^{8}} x^{8} - e^{- x^{8}}) + C

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What is the integral of 3x^{15}e^{-x^8} ?
The integral of 3x^{15}e^{-x^8} is 3/8 (-e^{-x^8}x^8-e^{-x^8})+C