00:01
So let's say we have the integral of sine x times cosine x dx.
00:11
We can solve this by using the u substitution u equals sine x, which means that d u equals cosine x d x.
00:26
Using this u substitution, we get the integral is just u, d, u.
00:34
If we use the power rule, this just becomes u squared over 2 plus c.
00:41
Then all we have to do is substitute sine x back in for u.
00:44
So we get one half times sine squared of x plus c.
00:52
So that's one answer we can get.
00:54
But what if we did the same integral, sine x, cosine, x, dx? but this time, we used the identity, sine x cosine x equals one -half sine 2x.
01:23
Now, if we use this identity, then we get something completely different.
01:28
So we can take the one -half outside the integral because it's constant.
01:35
Times sine 2x, dx.
01:41
From there, we can use the, we can use a u substitution, a fairly simple one where we say u equals 2x, so du equals 2dx.
01:56
And if we insert a two inside the integral, such that we can substitute for du, we then have to take out another one -half to keep things balanced.
02:08
So this becomes one -fourth because one -half times one -half outside the integral, times the integral of sine u -d -u, this becomes one -fourth sign or one -fourth times negative cosine of u plus c.
02:35
And if we substitute 2x back in for you, we get negative 1 4th times cosine 2x plus c.
02:46
So we got two answers using two different methods.
02:49
Negative 1 4th cosine 2x plus c versus 1 1 1⁄2x plus c versus 1 1 half sine squared x plus c.
03:07
And the reason for that is because while these two integrals are not exactly, equal, they're only different by a constant...