Integration by parts/substitution

[Nick Jarvis]

Hi. I think I jumped too eagerly into my latest module, as I am positive it makes many assumptions on previous experience. I am obviously lacking.

I have created a question in Word, but the equations don;t appear to be copying into this thread. So I have uploaded question instead. If this is against the rules please let me know, and I will attempt to re-create the equations here instead.

The question involves integration of ∫x(^(a-1)) e(^(-bx)) dx - and I have shown in the word doc what I think I need to do. However, the final answer is:

Γ(a)/ba

Which I know I won;t get from integration, but my answer should relate to this answer, using one of the distribution methods.

Many thanks

 

[BvU]

Hi,

Make life easy for yourself and study how to apply partial integration on $\displaystyle \int_0^\infty x^n e^{-x} dx$

(and yes, the Gamma function is involved)

 

[Nick Jarvis]

Thanks BvU, will take a look this evening.

 

[Nick Jarvis]

ok. I have integrated ∫xⁿ e^x using parts and I get:

xⁿ e^x - ∫e^xnx^n-1 this equates (I hope) to:

e^x(xⁿ - nx^n-1) this can be simplified obvs. Is this correct?

Thanks

 

[Nick Jarvis]

However, if I have ∫Bx^B-1 e^-xB dx, can I not integrate by substitution?

Let u = x^B so du = Bx^B-1dx which gives:

∫e^-udu and therefore the answer is -e^-xB + c

I know that is not correct, but I thought I could integrate by either substitution OR parts? This integration is key to me understanding stats/probability, but it has been many years since I have integrated.

Just looked again, and I know that ∫Bx^B-1 = x^B, but how do I incorporate that into it when I also have e_x^B

Many thanks

 

[BvU]

and I get: xⁿ e^x - ∫e^xnx^n-1 dx

correct so far.

this equates (I hope) to: e^x(xⁿ - nx^n-1)

Your hope is in vain. You can not just make an $\displaystyle\int$ sign disappear !

Do not confuse yourself by omitting the $dx$.

What are the bounds of the integral ? What did you do with them ? What is the meaning of the (correct) expression you got from integration by parts ? $\quad$ [edit] of the wrong expression $xe^x$ instead of $xe^{-x}$

 

[BvU]

Oops, I overlooked that you overlooked the minus sign in $e^{-x}$. Try again !

 

[Nick Jarvis]

Apologies. Have spoken also to my tutor and I have now worked it out. Thanks for your input. Much appreciated :)