Question

How to evaluate the integral using integration by parts with the indicated choices of u and dv. `int (lny)/sqrt(y) dy` ?

Answer 1

Refer to explanation

Explanation:

Using integration by parts we have

#int lny/sqrtydy=int 2*lny*(sqrty)'dy=2lny*sqrty-2int sqrty*(lny)'dy= 2lnysqrty-2int sqrty*(1/y)dy=2lnysqrty-2int(1/sqrty)=2lnysqrty-4*int (sqrty)'dy=2lnysqrty-4sqrty=2sqrty(lny-2)#

Hence finally we get

`int lny/sqrtydy=2sqrty*(lny-2)`

Remarks
1. For two functions f(x),g(x) integration by parts is

`int f(x)*(g(x))'dx=f(x)*g(x)-int g(x) * (f(x))' dx`

1. The phrase `()'` donates first derivative