Integrating $\int_0^{\sqrt{6}}e^{-x^2}\frac{x^2}{2}$: U-Substitution or Parts?

[RadiationX]

\int_0^\sqrt{6}}e^{-x^2}\frac{x^2}{2}

should i use a u-substitution or integration by parts?

 

[Zurtex]

You can not solve this in terms of elementary functions. It is possible to express the answer in terms of the Error function, that is:

\text{Erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt

You might want to give a shot at it yourself knowing that?

 

[RadiationX]

I don't think that I was being explicit enough in my first post: This is the double integral\int_0^3\int_{\sqrt{2y}}^{\sqrt{6}}e^{-x^2}dxdy that reduces to my original post integral

\int_0^\sqrt{6}}e^{-x^2}\frac{x^2}{2}dx

If I reduced correctly then how can i possibly integrate the integral with elementary functions?

 

[RadiationX]

Did I at least change the order of integration correctly?

 

[Zurtex]

Yes, you have changed it correctly.

It is not possible to integrate with elementary functions no matter how you look at it.

 

[Orion1]

\int \left( e^{-x^2} \frac{x^2}{2} \right) dx = \frac{1}{4} \left( \int_0^x e^{t^2} dt - e^{-x^2} x \right) + C

 

[kant]

\int_0^\sqrt{6}}e^{-x^2}\frac{x^2}{2}

should i use a u-substitution or integration by parts?

These a-hole intergral disgust me greatly when i worked on calaulus..

It is better if you just express e^t as a infinite serie. Substitude t=-x^2
in to the series. After that, multiple the entire series by x/2. intergrat it term by term, and plug numbers. This function can only be tame;not solve.

 

[lurflurf]

These a-hole intergral disgust me greatly when i worked on calaulus..

It is better if you just express e^t as a infinite serie. Substitude t=-x^2
in to the series. After that, multiple the entire series by x/2. intergrat it term by term, and plug numbers. This function can only be tame;not solve.

An answer in terms of erf is no worse than one in terms of sin or exp. There are table and computer programs to find values. Infinite series are helpful for some purposes, but unless one is going to compute an approximation by hand, an expression in terms of erf looks nicer and is more informative. Were you also discusted by integrals like
\int_1^2\frac{dx}{x}
What good is an answer like log(2) or sin(exp(sqrt(2))) anyway.
End special treatment for elementary function.
Equality for special functions.
Equal rights for all functions.

 

[TenaliRaman]

End special treatment for elementary function.
Equality for special functions.
Equal rights for all functions.

Mathematical constipation
I mean constitution

-- AI

 

[Zurtex]

What good is an answer like log(2) or sin(exp(sqrt(2))) anyway.
End special treatment for elementary function.
Equality for special functions.
Equal rights for all functions.

Well in that case my new function is called easyanswer(t), easyanswer(t) is defined such that where t is some real number of my choice it is the solution to the given numerical integral in front of me. Much easier exams now