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Popular >

integral of x^33^{-x}

Solution

\int x^{3} 3^{- x} d x

Solution

- \frac{3 ^{- x} x ^{3}}{ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{3 ^{- x + 1} x ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (- \frac{3 ^{- x + 1} x}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 - x})) + C

Solution steps

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

Apply u-substitution

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

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

Apply Integration By Parts

: \frac{3 ^{u} u ^{3}}{ln ( 3 )} - \int \frac{3 ^{u + 1} u ^{2}}{ln ( 3 )} d u

= \frac{3 ^{u} u ^{3}}{ln ( 3 )} - \int \frac{3 ^{u + 1} u ^{2}}{ln ( 3 )} d u

\int \frac{3 ^{u + 1} u ^{2}}{ln ( 3 )} d u = \frac{1}{ln ( 3 )} (\frac{3 ^{u + 1} u ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (\frac{3 ^{u + 1} u}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 + u}))

= \frac{3 ^{u} u ^{3}}{ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{3 ^{u + 1} u ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (\frac{3 ^{u + 1} u}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 + u}))

Substitute back

u = - x

= \frac{3 ^{- x} ( - x ) ^{3}}{ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{3 ^{- x + 1} ( - x ) ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (\frac{3 ^{- x + 1} ( - x )}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 - x}))

Simplify

\frac{3 ^{- x} ( - x ) ^{3}}{ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{3 ^{- x + 1} ( - x ) ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (\frac{3 ^{- x + 1} ( - x )}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 - x})) : - \frac{3 ^{- x} x ^{3}}{ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{3 ^{- x + 1} x ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (- \frac{3 ^{- x + 1} x}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 - x}))

= - \frac{3 ^{- x} x ^{3}}{ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{3 ^{- x + 1} x ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (- \frac{3 ^{- x + 1} x}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 - x}))

Add a constant to the solution

= - \frac{3 ^{- x} x ^{3}}{ln ( 3 )} - \frac{1}{ln ( 3 )} (\frac{3 ^{- x + 1} x ^{2}}{ln ( 3 )} - \frac{2}{ln ( 3 )} (- \frac{3 ^{- x + 1} x}{ln ( 3 )} - \frac{1}{ln ^{2} ( 3 )} \cdot 3^{1 - x})) + C

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Frequently Asked Questions (FAQ)

What is the integral of x^33^{-x} ?
The integral of x^33^{-x} is