Basic Integration By Parts

This is a post based on a question.

If it isn't a problem, can I ask you how to integrate by parts?

The properties of integrals let us split up addition, but not multiplication.

So, when we have two functions MULTIPLIED together, we must use the technique of Integration By Parts.

(Note: In this post I will show the formula and explain how to use it, but I will not show why it works. If anyone would like to see where the formula comes from, please let me know in a comment.)

Notice that there is an integral in the answer! This means that to evaluate the problem completely you must also do that integral at the end. This also means that there is potential for the process repeating over and over (you get the answer, but then you have to use Integration By Parts again), but I titled this post Basic Integration By Parts for a reason; repetition would be Advanced Integration By Parts.

Anyway, this formula is critical. After extensive practice, you will know it by heart just because you will use it so much. Before I was entirely familiar with it, however, I had a chant that helped me remember it. All I said was "u v minus v du," but it was enough for me remember it forever.

Now, let's do an example.
Actually, more like continue the example I began working with. For this first one, I will choose which function is u and which function is dv just so the process is clear. Later, I will teach you how to choose u and dv for yourself (which is the hardest step in Integration By Parts).

So we have all of the pieces. Now we must organize them and put them into the formula.

Now we should check our work by taking the derivative of our answer and seeing if it matches the original question.

Remember that this technique is used when two functions are being multiplied. But since they're being multiplied, it could be in any order. For example, x * cos x could have been written as cos x * x. So how do you figure out which one is u and which one is dv? Mostly, through practice. It only works one way, so if you try it and it doesn't work out, simply switch them.

The practice required to start immediately recognizing u and dv without mistakes is important, but also time consuming. Luckily, like most things in math, I have a trick for this. Unlike most things in math, I actually have 2 tricks.

The first trick is the way I personally learned, and the concept behind it is important enough that I put it in a specially-colored box.

The second trick is something I picked up from a teacher.  All you have to remember is "LIATE". Pick your u by whatever is first/highest in the continuum below. Or, backwards, you could pick your dv by whatever comes lowest in the continuum.

It's time for another example of Integration By Parts.

This technique is very difficult because it has many small details that must be accounted for, in addition to the challenge of the actual calculus and algebra involved.
Any given calculus textbook probably has 50 or so if these problems. The only way to truly learn this method is practice it over and over again.