Integration by parts with substitution

Integration by parts I understand, but part of the problems for this section has instructions to "First make a substitution and then use integration by parts to evaluate the integral". The book itself doesn't even mention this in the section. I have the answers, but it's not really helping me understand what's going on or why.

example:

The answer says to substitute for , so

.

Thus

Switching in the y is a bit of a stretch. I understand it, but I'm not sure I understand why. The book then shows to substitute all this back into the original problem to get

I don't understand this at all... how'd we get from to ? From there, it's just integration by parts, and I understand the process. It's only the substitution part that's got me baffled.

If someone is able to explain the how and why of that, I'd be much appreciative.

Re: Integration by parts with substitution

If you have:

Let

So the integral can be written as:

Note:

If you use the variable then it's confusing to suddenly use the variable so stay with one variable.

Re: Integration by parts with substitution

Quote:

Originally Posted by

**satis** Integration by parts I understand, but part of the problems for this section has instructions to "First make a substitution and then use integration by parts to evaluate the integral". The book itself doesn't even mention this in the section. I have the answers, but it's not really helping me understand what's going on or why.

example:

The answer says to substitute for

, so

.

Thus

This is completely wrong because you forgot the "dx" when you differentiated. You should have

from that, or .

Quote:

Switching in the y is a bit of a stretch. I understand it, but I'm not sure I understand why. The book then shows to substitute all this back into the original problem to get

I don't understand this at all... how'd we get from

to

? From there, it's just integration by parts, and I understand the process. It's only the substitution part that's got me baffled.

If someone is able to explain the how and why of that, I'd be much appreciative.