00:01
Let's evaluate the indefinite integral of x to the power 7 times cosine of x to the power 4.
00:07
We might attempt integration by parts on this, however, with the powers on 7 for x and 4 on x inside cosine, that doesn't look like it'll be a very fruitful method to pursue.
00:18
Instead, what we're going to do is try to come up with a u substitution that can simplify this integrand.
00:25
Let's try with u as equal to x power 4.
00:28
That would give us a cosine u which is easier to work with and with this substitution the differential du would be equal to 4 times x power 3 dx or if we prefer a fourth du would be equal to x cubed d x let's go back to the statement of the original problem and reorganize it we have x power 7 but we have to give up x cubed in this solution.
00:57
So i'm going to factor as x power 4 times cosine of x power 4 times x cubed dx.
01:08
Now notice at this stage we're in business because x cubed dx is replaceable by a fourth du and now we have x power 4 here and here which is replaceable by the quantity you.
01:23
So with those two substitutions, we'll obtain anti -derivative of x4 became u times cosine of x4, which is you again, times x -cube dx, which is a fourth du.
01:39
I'll place du here and one -fourth outside of the integral.
01:44
At this stage, u times cosine u can be handled by the method of integration by parts.
01:50
Let's let w be equal to u so that the differential dw equals simply du...