[MUSIC] Integrating requires patience. 
Sometimes you can start attacking the 
monster, but it might take a little bit 
of time before you start to see success. 
Prepare to persevere. 
Let's attack the indefinite integral of 
cosx times e to the x dx. 
There's a metaphor here. 
There's really different ways to win the 
battle. 
Suppose this is you, and this is the 
integral that you're trying to attack. 
One way, is you know, to try and do it by 
hand, right. 
To take your shield and to take your 
sword, and just to evaluate that integral 
directly by, say going back to the 
definition of Riemann sum. 
But we almost never attack integrals in 
this way. 
Instead of using, you know, swords and 
shields, we're really using, you know, 
magic. 
Right? 
We're putting on our wizard hat. 
And instead of trying to kill the 
intregral, right? 
We use something like u substitution to 
transform that integration problem into a 
friendly integration problem that we can 
actually deal with, right? 
We're never attacking integrals directly 
we're just transforming integrals by 
doing u substitutions or parts. 
So let's try to transform the integration 
that we're attacking here. 
Let's tranform e to the x cosx to 
something else. 
So I'll use what u, I'll make u be e to 
the x. 
because I don't really mind 
differentiating that. 
And I'll make dv cosx dx. 
Now in that case, the derivative of e to 
the x is just e to the x. 
And what's an anti-derivative of cosine? 
An anti-derivative of cosine is sine. 
Now what is integration by parts say. 
So then anti-derivative of cosx e to the 
x dx is U times V, so e to the x sinx 
minus the intergral of V du So, sinx, e 
to the x dx. 
Well that didn't really help matters. 
But the word of the day is persevere. 
So let's persevere, and transform that 
integral into yet another monster. 
So the monster in this case is this thing 
here. 
let's try the same trick on that new 
integral. 
So U will be e to the x again and dv 
here, will be just what's left over, sine 
x dx. 
And in that case, if U is e to the x, 
then du is e to the x dx. 
And, if dv is sin x dx, well, I've got 
other choices for more anti-derivative, 
but I'll have v be minus cosx. 
Now, I've put this into the integration 
by parts formula. 
So the antiderivative of sinx e to the x 
dx is uv, so minus e to the x cosx minus 
vdu. 
So my-, the integral of vdu, so v is 
cosx, and du is e to the x dx. 
Now what's happened thus far in the 
battle? 
I'm attacking this integral. 
I used integration by parts to transform 
the monster into something else. 
It didn't help, so I used that magical 
weapon again. 
I used integration by parts again to 
transform the integration problem into 
yet another integration problem and it 
didn't help. 
Or did it? 
What happens if I put these two 
integration facts together? 
Well, and then I've got here, the 
integral of cosx, e to the x dx, is equal 
to e to the x sin x minus the integral of 
sinx, e to the x, dx, which is down here. 
So minus negative e to x cos x plus the 
integral of cosine x e to the dx. 
Lets persevere, lets keep going. 
So this is saying the integral of cos x e 
to the x dx is e to the x sin x plus e to 
the x, cosine x, minus the integral of 
cosine x, e to the x, dx. 
Now I'm going to add the integral of cos 
x e to the x, dx to both sides. 
And what I'll then get is 2 times the 
integral of cos x e to the x, dx is equal 
to, well what's left on the righthand 
side, e to the x sin x plus e to the x, 
cos x. 
And then I'll divide both sides by 2, and 
if I divide this side by 2 and this side 
by 2 well then these 2's cancel, and I 
find out that an antiderivative of cosine 
x e to the x Is e to the x sin x plus z 
to the x cos x all over 2. 
What we did here is one of my favorite 
applications of integration by parts. 
We didn't use integration by parts to 
solve, to evaluate the integral directly. 
Instead, I applied parts twice and ended 
up with a clone of the original monster. 
So how did that go? 
Well, here you are, right, and you're a 
wizard, so here's your wizard hat, and 
here's the integral that we wanted to 
attack. 
But we didn't attack this integral 
directly. 
We didn't' even transform this integral 
by using U substitution. 
Instead, we used parts twice and it 
seemed like that was a terrible idea 
because when I applied parts twice, I 
ended up with a clone of the original 
monster. 
The same integral ended up appearing. 
But, that other integral came with a 
minus sign. 
And, consequently that monster ends up 
defeating itself and we're able to 
evaluate the integral without actually 
integrating at all. 
If anything suggests that there's 
creativity in the battles ahead of us, I 
think integration by parts is telling us, 
to be prepared to be amazed.