Deduce an integral I came across

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In summary, the conversation is about a person trying to solve an integral involving x^3e^{-x^2}. They have tried partial integration and substitution, but have not been successful. Another person suggests using the substitution u=x^2 and integrating by parts to solve the integral.
  • #1
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Hey guys, I'm trying to deduce an integral I came across whilst studying some thermodynamics, but I can't seem to evaluate it:
[tex]\int_0^\infty x^3e^{-x^2}dx[/tex]
I've tried partial integration numerous times, but I can't seem to get it right. Can you help me?
I consider
[tex]\int_{-\infty}^\infty e^{-x^2}dx = \frac{1}{2}\sqrt{\pi}[/tex]
as a standard integral.
 
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  • #2


Try substitution [tex]u=x^2[/tex].
 
  • #3


Thanks, I did try that substitution before, but now I tried it again and it came out, I must have made a mistake. Thanks!
 
  • #4


Specically, write the integral as
[tex]\int_0^\infty x^2e^{-x^2}(x dx)[/tex]
and then set [itex]u= x^2[/itex]. You will get an integral of the form
[tex]\frac{1}{2}\int_0^\infty ue^{-u}du[/tex]
that you can integrate by parts.
 

FAQ: Deduce an integral I came across

What is an integral?

An integral is a mathematical concept that is used to find the area under a curve in a graph. It is also known as the anti-derivative of a function. It is represented by the symbol ∫ and has a lower and upper bound.

How do I deduce an integral?

To deduce an integral, you must use the fundamental theorem of calculus which states that the integral of a function can be calculated by finding the anti-derivative of the function and evaluating it at the upper and lower bounds. This process is also known as integration by substitution.

What are the different types of integrals?

There are two main types of integrals: definite and indefinite. A definite integral has specific bounds and gives a numerical value, while an indefinite integral has no bounds and gives a general formula. Other types of integrals include improper integrals and line integrals.

What are some common techniques used to deduce an integral?

Some common techniques used to deduce an integral include integration by parts, trigonometric substitution, partial fractions, and u-substitution. These techniques involve manipulating the integrand to make it easier to integrate and finding the anti-derivative.

How can I check if my deduced integral is correct?

You can check if your deduced integral is correct by using the properties of integrals, such as linearity and the fundamental theorem of calculus. You can also use online calculators or software to verify your integral. Additionally, you can differentiate your integral and see if it gives you the original function.

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