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Popular >

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

= - \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

Popular Examples

y^{''}+y=0,y(pi/3)=8,y^'(pi/3)=-16

y^{^{''}} + y = 0, y (\frac{π}{3}) = 8, y^{^{'}} (\frac{π}{3}) = - 16

inverse oflaplace (s+10)/(s^3(s+2.5)^2)inverse laplace

\frac{s + 10}{s ^{3} ( s + 2.5 ) ^{2}}

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\int_{a}^{p} \frac{b}{x ( 1 - x )} d x

derivative of \sqrt[6]{x^5}

Frequently Asked Questions (FAQ)

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