A) γ(n) =∫[infinity]0tn−1e−tdta)show that γ(n+ 1) =nγ(n) using integration by parts. b) show that γ(n+ 1) =n! , wherenis a positive integer, by repeatedly applying the result in parta)c) compute γ(5).d) prove what the value of γ(12) should be by evaluating the gamma function forn= 1/2 and usingthe fact that∫[infinity]0e−u2du=√π2.

a. Integrate by parts by taking

Then

We have

and so

or, replacing , .

b. From the above recursive relation, we find

Now,

and so we're left with .

c. Using the previous result, we find .

d. If the question is asking to find , then you can just use the same approach as in (c).

But if you're supposed to find , we have

Substitute

Then

15 batches are the most you can make

step-by-step explanation:

look at the picture i have filled out a table for it. hope this .