00:01
Okay, we want to find the indefinite integral of sine x, cosine x times natural log of sine of x.
00:08
And to do this, we'll be using, again, u substitution in a way that allows us to use a specific solution found on an integral table.
00:18
So for this, we're going to say that u is equal to sine of x, which in turn means u squared is sine square of x.
00:24
And if we do the derivative of our u equals sinex, we get that du is equal to cosine x d x, d x.
00:31
And this u sub works out pretty well because we already have a cosine dx cosine dx in our original function so u substitution goes well without having any other constants or other factors to worry about so this means that our integral is now the integral of u squared times natural log of u d u so for this there is a specific solution to this by the use of integral table and this is the fact that given the following function, u to the n times natural log of u, du, you get the following solution, which is u to the n plus 1 over n plus 1 squared times in parentheses n plus 1 times natural log of u quantity minus 1 plus c...