00:01
So we want to find the following integral using integration by parts.
00:06
So first let's rewrite this integral a little bit nicer.
00:11
So this exponent means we have t squared times lnt squared d t.
00:22
So for our integration by parts, let's let you be equal to lnt squared.
00:39
And let's let v be equal to t squared d t.
00:44
So now differentiating we get d u is equal to 2 times lntt times the chain rule, which is 1 over t d t.
01:00
And sorry this should be dv up here.
01:05
So v after we integrate is going to be 1 3t cubed.
01:15
So now we can rewrite our integral.
01:20
So using integration by parts here, we have uv, which is 1 3rd t cubed times l &t squared minus.
01:36
Now we need to integrate v, du.
01:40
So v and du, we have some constants here.
01:45
So 2 times a third is 2 thirds, and we'll just go ahead and stick it out front.
01:49
And what we have inside is lnt, and we also have t cubed over t, which gives us a t squared, dt.
02:02
So now we need to worry about this integral right here...