What is the integral of 7x^{11}e^{-x^6} ?

The integral of 7x^{11}e^{-x^6} is 7/6 (-e^{-x^6}x^6-e^{-x^6})+C

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integral of 7x^{11}e^{-x^6}

\int 7 x^{11} e^{- x^{6}} d x

\frac{7}{6} (- e^{- x^{6}} x^{6} - e^{- x^{6}}) + C

Solution steps

\int 7 x^{11} e^{- x^{6}} d x

Take the constant out:

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

= 7 \cdot \int x^{11} e^{- x^{6}} d x

Apply u-substitution

= 7 \cdot \int \frac{e ^{u} u}{6} d u

Take the constant out:

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

= 7 \cdot \frac{1}{6} \cdot \int e^{u} u d u

Apply Integration By Parts

= 7 \cdot \frac{1}{6} (e^{u} u - \int e^{u} d u)

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

= 7 \cdot \frac{1}{6} (e^{u} u - e^{u})

Substitute back

u = - x^{6}

= 7 \cdot \frac{1}{6} (e^{- x^{6}} (- x^{6}) - e^{- x^{6}})

Simplify

7 \cdot \frac{1}{6} (e^{- x^{6}} (- x^{6}) - e^{- x^{6}}) : \frac{7}{6} (- e^{- x^{6}} x^{6} - e^{- x^{6}})

= \frac{7}{6} (- e^{- x^{6}} x^{6} - e^{- x^{6}})

Add a constant to the solution

= \frac{7}{6} (- e^{- x^{6}} x^{6} - e^{- x^{6}}) + C

What is the integral of 7x^{11}e^{-x^6} ?
The integral of 7x^{11}e^{-x^6} is 7/6 (-e^{-x^6}x^6-e^{-x^6})+C