I with finding the intergral of (x^n)(e^x)

[minase]

I need help with finding the intergral of (x^n)(e^x). I already did for 1,2 and 3. I found out that x^n*e^x times the intergral of (x^n-1)e^x. Is there anyway I can genralize for all n s without having an integral.

 

[dontdisturbmycircles]

Post this in the homework section "Calculus and Beyond" and you are much more likely to get an answer.

 

[chanvincent]

Easy, use integration by part, show the following relationship
$$\int x^n e^x dx = x^ne^x -n\int x^{n-1}e^x dx$$

Then use the recurrence relation on $$\int x^{n-1} e^x dx$$.
You will get
$$\int x^{n} e^x} dx = \sum _{i=0} ^{n} (-1)^i\frac{n!}{(n-i)!}e^xx^{n-i}$$

 

[Gib Z]

chanvincent is correct. when its the same integral with e^NEGATIVE x, its a nice definition for the gamma function.