What is the integral of x^8(3^{-x^9}) ?

The integral of x^8(3^{-x^9}) is -(3^{-x^9-2})/(ln(3))+C

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integral of x^8(3^{-x^9})

\int x^{8} (3^{- x^{9}}) d x

- \frac{3 ^{- x^{9} - 2}}{ln ( 3 )} + C

Solution steps

\int x^{8} \cdot 3^{- x^{9}} d x

Apply u-substitution

: \int - 3^{u - 2} d u

= \int - 3^{u - 2} d u

Take the constant out:

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

= - \int 3^{u - 2} d u

Apply u-substitution

: \int 3^{v} d v

= - \int 3^{v} d v

\int a^{x} d x = \frac{a ^{x}}{ln a}

= - \frac{3 ^{v}}{ln ( 3 )}

Substitute back

= - \frac{3 ^{- x^{9} - 2}}{ln ( 3 )}

Add a constant to the solution

= - \frac{3 ^{- x^{9} - 2}}{ln ( 3 )} + C

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What is the integral of x^8(3^{-x^9}) ?
The integral of x^8(3^{-x^9}) is -(3^{-x^9-2})/(ln(3))+C