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Integrate the function $x^3e^{-2x}$ from 0 to $\infty $

Used Formulas

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asin
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sinh
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asinh
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Basic Derivatives

· Power rule for derivatives
$\frac{d}{dx}\left(x^a\right)=ax^{\left(a-1\right)}$
$\frac{d}{dx}\left(cx\right)=c\frac{d}{dx}\left(x\right)$
· Derivative of the linear function
$\frac{d}{dx}\left(x\right)=1$
· Derivative of a Constant
$\frac{d}{dx}\left(c\right)=0$

Integration Techniques

· Integration by Substitution
$\int f\left(x\right)dx=\int f\left(g\left(t\right)\right) g'\left(t\right)dt$

Basic Integrals

$\int e^xdx=e^x+C$
· Constant factor Rule
$\int cxdx=c\int xdx$

Function Plot

Plotting: $x^3e^{-2x}$

Main Topic: Definite Integrals

Given a function f(x) and the interval [a,b], the definite integral is equal to the area that is bounded by the graph of f(x), the x-axis and the vertical lines x=a and x=b

Used Formulas

7. See formulas

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