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Integrate the function $x^2e^{-x}$ from 0 to $\infty $

Step-by-step Solution

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asinh
acosh
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Final Answer

$2$
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Step-by-step Solution

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We can solve the integral $\int x^2e^{-x}dx$ by applying the method of tabular integration by parts, which allows us to perform successive integrations by parts on integrals of the form $\int P(x)T(x) dx$. $P(x)$ is typically a polynomial function and $T(x)$ is a transcendent function such as $\sin(x)$, $\cos(x)$ and $e^x$. The first step is to choose functions $P(x)$ and $T(x)$

$\begin{matrix}P(x)=x^2 \\ T(x)=e^{-x}\end{matrix}$

Learn how to solve differential calculus problems step by step online.

$\begin{matrix}P(x)=x^2 \\ T(x)=e^{-x}\end{matrix}$

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Learn how to solve differential calculus problems step by step online. Integrate the function x^2e^(-x) from 0 to infinity. We can solve the integral \int x^2e^{-x}dx by applying the method of tabular integration by parts, which allows us to perform successive integrations by parts on integrals of the form \int P(x)T(x) dx. P(x) is typically a polynomial function and T(x) is a transcendent function such as \sin(x), \cos(x) and e^x. The first step is to choose functions P(x) and T(x). Derive P(x) until it becomes 0. Integrate T(x) as many times as we have had to derive P(x), so we must integrate e^{-x} a total of 3 times. With the derivatives and integrals of both functions we build the following table.

Final Answer

$2$

Exact Numeric Answer

$2$

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Function Plot

Plotting: $x^2e^{-x}$

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1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

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