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Integration by Parts

IMG_2780.jpg

Can someone help with this? No idea how to do the first part. I've attached my working out so far.

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Reply 1
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Notnek
21
Original post by hatedigi


Can someone help with this? No idea how to do the first part. I've attached my working out so far.


You seem to be saying that the derivative wrt t of txt^x is 1 but I'm not sure why?
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hatedigi
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Original post by Notnek
You seem to be saying that the derivative wrt t of txt^x is 1 but I'm not sure why?


nvm, how would I differentiate it with respect to t then?
(edited 6 years ago)
Reply 3
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Notnek
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Original post by hatedigi
nvm, how would I differentiate it with respect to t then?

Treat xx as a constant so you can differentiate it just like you would differentiate t3t^3 for example.
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hatedigi
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Original post by Notnek
Treat xx as a constant so you can differentiate it just like you would differentiate t3t^3 for example.


Ok so this is what I have now but I have an extra - e ^ -t * t^ ximage.jpg
Reply 5
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Notnek
21
Original post by JustChilling24/7


Maybe this will help for the first part

I had to remove your post unfortunately because it contained a full solution which is against the maths forum rules. It would be great if you can help the OP at the point they are stuck without giving a solution.

I won't be able to help until later.
Original post by Notnek
I had to remove your post unfortunately because it contained a full solution which is against the maths forum rules. It would be great if you can help the OP at the point they are stuck without giving a solution.

I won't be able to help until later.


Oops I did not know that sorry my bad.
Try to integrate gamma(x) instead of gamma(x+1)
Original post by hatedigi
Ok so this is what I have now but I have an extra - e ^ -t * t^ ximage.jpg
Reply 8
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hatedigi
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Original post by JustChilling24/7
Try to integrate gamma(x) instead of gamma(x+1)


Still got a term at the front getting in the way unfortunately image.jpg
Original post by hatedigi
...


When using integration by parts on a definite integral you need to evaluate the term you've separated out at the two limits of the integral.


e−ttxe^{-t}t^x

should be more correctly written as:

[e−ttx]0∞\displaystyle \left[e^{-t}t^x\right]_0^{\infty}
You need to evaluate your [e-t tx] between infinity and zero (this is integration by parts). You should notice something happen to this expression when you sub in the limits.
Original post by hatedigi
Still got a term at the front getting in the way unfortunately image.jpg
It seems like you're struggling to deal with infinite limits, as they aren't taught at A level.
∫0∞F(t)dt=limu→∞[f(t)]0u \displaystyle \int_{0}^{\infty}{F(t)}dt = lim_{u \rightarrow \infty} [f(t)]_{0}^{u} , where F(t) is the derivative of f(t), in this case the limit exists so there will be a convergence; just consider what factor grows faster?

I remember finding a good explanation of this if you Google "Paul's math notes".

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