I understand that e^(x²) does not have an elementary derivative and the calculations to get that as a definite integral with limits infinity and negative infinity, the value obtained is √π. In my latest attempt at this, I replaced e^(x²) with e^(u²) from the original equation (since substitution was requested in the question) so that I wouldn't get confused and after getting an equation I don't think I can work with, I'm not sure if I'm approaching this question correctly at all.

Can someone help to point me in the right direction? Thanks

Can someone help to point me in the right direction? Thanks

(edited 3 weeks ago)

Original post by SCQ

I understand that e^(x²) does not have an elementary derivative and the calculations to get that as a definite integral with limits infinity and negative infinity, the value obtained is √π. In my latest attempt at this, I replaced e^(x²) with e^(u²) from the original equation (since substitution was requested in the question) so that I wouldn't get confused and after getting an equation I don't think I can work with, I'm not sure if I'm approaching this question correctly at all.

Can someone help to point me in the right direction? Thanks

Can someone help to point me in the right direction? Thanks

u = (x-mu)/sqrt(2)sigma

should be straightforward

Original post by mqb2766

u = (x-mu)/sqrt(2)sigma

should be straightforward

should be straightforward

So I tried this earlier and found dx in terms of du:

u = x/(√2σ) - μ/(√2σ)

du/dx = 1/(√2σ)

dx = (√2σ)du

And substituting this gave me: ∫(e^(u²))(√2σ)du, and I’m not sure where to go from there

Original post by ssccqq

So I tried this earlier and found dx in terms of du:

u = x/(√2σ) - μ/(√2σ)

du/dx = 1/(√2σ)

dx = (√2σ)du

And substituting this gave me: ∫(e^(u²))(√2σ)du, and I’m not sure where to go from there

u = x/(√2σ) - μ/(√2σ)

du/dx = 1/(√2σ)

dx = (√2σ)du

And substituting this gave me: ∫(e^(u²))(√2σ)du, and I’m not sure where to go from there

Its e^(-u^2) and your definite integral limits should be similar and sqrt(2) and sigma are constants so its pretty much a write down answer

(edited 3 weeks ago)

Original post by mqb2766

Its e^(-u^2) and your definite integral limits should be similar and sqrt(2) and sigma are constants so its pretty much a write down answer

Sorry, you’re right, it’s -u²! I think I might be overcomplicating this, so I’m also really sorry for sounding stupid right now, but am I just supposed to compare with the original equation (∫(e^(-u²)du=√π) and add in the coefficient to make it fit?

Original post by ssccqq

Sorry, you’re right, it’s -u²! I think I might be overcomplicating this, so I’m also really sorry for sounding stupid right now, but am I just supposed to compare with the original equation (∫(e^(-u²)du=√π) and add in the coefficient to make it fit?

Sure youve a constant multiplier so take it outside the integral and im presuming youve worked out that the limits dont chage

(edited 3 weeks ago)

Original post by mqb2766

Sure youve a constant multiplier so take it outside the integral and im presuming youve worked out that the limits dont chage

Ah, thank you so much for your help! I’m so sorry for bugging you with this; I thought I had to find values for σ and μ (even though there’s only one equation, so pretty sure that’d be impossible) or something more complicated. Yeah, really straightforward, haha

Original post by ssccqq

Ah, thank you so much for your help! I’m so sorry for bugging you with this; I thought I had to find values for σ and μ (even though there’s only one equation, so pretty sure that’d be impossible) or something more complicated. Yeah, really straightforward, haha

Not sure of the context of the question, but Im sure you recognise that youre basically working out the normalising factor for a normal distribution. The mean mu doesnt affect things as it just shifts the location of the bump and as youre integrating between -inf and inf, thats irrelevant. However the scaling term 1/sqrt(2)sigma must affect the width and so the value of the integral.

Original post by mqb2766

Not sure of the context of the question, but Im sure you recognise that youre basically working out the normalising factor for a normal distribution. The mean mu doesnt affect things as it just shifts the location of the bump and as youre integrating between -inf and inf, thats irrelevant. However the scaling term 1/sqrt(2)sigma must affect the width and so the value of the integral.

I did find that picking variables σ and μ reminded me of normal distribution, but I couldn’t remember the specific equation. Interesting to know! This is just a standalone question for pre-uni revision, but I’ll definitely add normal distribution to the long list of what to read up on before uni

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