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How do you integrate by parts?

This is one of the trickiest methods of calculus on the course, but it's important to know, and is very doable if you set up the problem right and remember the steps. 

Integration by parts works when you have to integrate a function of the type f=u(dv/dx). All you have to remember is that, and the formula(dv/dx) dx = uv - ∫ (du/dx) dx

Ok, let's try an example. 

Say you're asked to integrate xsin(x). 

I find it makes it easiest to write out all the things I need for the formula before I plug them in. 

We'll choose x to be u, because differentiating x makes it more simple, while differentiating sin(x) doesn't really help that much. You always choose u to be the part that comes out simplest when differentiated.

So:                                                            u = x

Then, by differentiating,                du/d= 1

and also:                                               dv/dx = sin(x)

Then, integrating to find v,             v = -cos(x). 

Now, all we have to do is plug that back into the formula from earlier:

             ∫ xsin(x) dx = -xcos(x) - ∫ -cos(x) (1) dx.

Which is way easier! Integrating cos(x) gives sin(x) + c (always remember c!), so we end up with

             ∫ xsin(x) dx = -xcos(x) + sin(x) + c.

And that's your answer!

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