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Step-by-step Solution

Integrate $x^2e^{-2x}$ with respect to x

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Answer

$-\frac{1}{2}e^{-2x}x^2-\frac{1}{2}e^{-2x}x-\frac{1}{4}e^{-2x}+C_0$

Step-by-step explanation

Problem to solve:

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

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

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$
2

First, identify $u$ and calculate $du$

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

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Answer

$-\frac{1}{2}e^{-2x}x^2-\frac{1}{2}e^{-2x}x-\frac{1}{4}e^{-2x}+C_0$
$\int x^2 e^{-2x}dx$

Main topic:

Integration by substitution

Used formulas:

5. See formulas

Time to solve it:

~ 0.8 seconds