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Popular Calculus >

integral of x^3*2^{-x}

Solution

\int x^{3} \cdot 2^{- x} d x

Solution

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

Solution steps

\int x^{3} \cdot 2^{- x} d x

Apply Integration By Parts

= - \frac{2 ^{- x} x ^{3}}{ln ( 2 )} - \int - \frac{3 \cdot 2 ^{- x} x ^{2}}{ln ( 2 )} d x

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

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

Add a constant to the solution

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

Graph

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Frequently Asked Questions (FAQ)

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The integral of x^3*2^{-x} is