What is the integral of (x^2)/(e^{x/3)} ?

The integral of (x^2)/(e^{x/3)} is -3e^{-x/3}x^2+6(-3e^{-x/3}x-9e^{-x/3})+C

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integral of (x^2)/(e^{x/3)}

\int \frac{x ^{2}}{e ^{\frac{x}{3}}} d x

- 3 e^{- \frac{x}{3}} x^{2} + 6 (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}}) + C

Solution steps

\int \frac{x ^{2}}{e ^{\frac{x}{3}}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

= \int x^{2} e^{- \frac{x}{3}} d x

Apply Integration By Parts

: - 3 e^{- \frac{x}{3}} x^{2} - \int - 6 e^{- \frac{x}{3}} x d x

= - 3 e^{- \frac{x}{3}} x^{2} - \int - 6 e^{- \frac{x}{3}} x d x

\int - 6 e^{- \frac{x}{3}} x d x = - 6 (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}})

= - 3 e^{- \frac{x}{3}} x^{2} - (- 6 (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}}))

Simplify

= - 3 e^{- \frac{x}{3}} x^{2} + 6 (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}})

Add a constant to the solution

= - 3 e^{- \frac{x}{3}} x^{2} + 6 (- 3 e^{- \frac{x}{3}} x - 9 e^{- \frac{x}{3}}) + C

5 \cdot t

\int \frac{1}{x ^{2} x ^{2} + 25} d x

\frac{\partial}{\partial y} (\frac{x}{y})

\frac{4083}{1 + 6.22 e ^{- 0.32 x}}

f (x) = \frac{4}{3} π r^{3}

What is the integral of (x^2)/(e^{x/3)} ?
The integral of (x^2)/(e^{x/3)} is -3e^{-x/3}x^2+6(-3e^{-x/3}x-9e^{-x/3})+C