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Step-by-step Solution

Integrate x^2e^(-3x)

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Answer

$-\frac{1}{3}e^{-3x}x^2+\frac{2}{3}\left(-\frac{1}{3}e^{-3x}x-\frac{1}{9}e^{-3x}\right)+C_0$

Step-by-step explanation

Problem to solve:

$\int\:x^2\:e^{-3x}dx$
1

Use the integration by parts theorem to calculate the integral $\int x^2e^{-3x}dx$, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
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First, identify $u$ and calculate $du$

$\begin{matrix}\displaystyle{u=x^2}\\ \displaystyle{du=2xdx}\end{matrix}$

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Answer

$-\frac{1}{3}e^{-3x}x^2+\frac{2}{3}\left(-\frac{1}{3}e^{-3x}x-\frac{1}{9}e^{-3x}\right)+C_0$

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$\int\:x^2\:e^{-3x}dx$

Main topic:

Integration by parts

Used formulas:

5. See formulas

Time to solve it:

~ 0.5 seconds