Integration by Parts — Lee Ziyang

Description: This FOTW issue presents a shortcut to using the integration by parts technique to integrate stuff. While this technique is taught in JC2 H2 math, I think using the shortcut presented makes life easier. In this FOTW, we’ll first see how integration by parts is done the normal way, before learning the shortcut. The stuff presented here might be relevant to H2 math.

Credits: Credit goes to Dr Phil Chan, a physics professor in NUS, who taught this technique to the 2008 Sec 3 Phy RA students as part of the physics evening lectures on Fourier transforms.

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Integration by parts is a very useful integration technique. It is derived from the product rule of differentiation.

Integrating:

Rearranging the terms:

The name and origin of his technique suggests that it should be used when you have to integrate a product. Without using the shortcut, let’s try integrating x2sinx . There are two terms here: x2 and sinx . From the formula, we know that we’ll have to integrate one of them, and differentiate one of them. The choice of which term to differentiate or integrate is entirely ours. Some experience will tell us that we should choose to differentiate the x2 term and integrate the sinx term:

The thing that we need to integrate is again a product of two terms. So we use integration by parts again. Be sure to be consistent in the terms you choose to differentiate and integrate. Since we chose to differentiate the x2 term just now, we should follow through differentiating it all the way. Thus,

It can get quite confusing. Got to integrate, then differentiate then integrate the product of the integrated term and differentiated term. Even the sentence above is confusing… Fortunately, the shortcut makes life easier. Here it is, illustrated using the example above:

Choose a term to differentiate and a term to integrate. In other words, choose something to be your u and something to be your  . Again, let’s choose our x2 to be the u, and sinx to be the . Then we do this:

* Differentiate the u term repeatedly.

* Integrate the  term repeatedly.

* Starting from the 2nd row, fill in the alternating signs “+, -, +, -…” repeatedly

*Multiply the terms as shown in the table above and factor in the relevant signs in the 3rd column.

You get this result:

Which is exactly the same as the result we get from doing it the tedious way. Whee. No integration signs needed at all. Why does it work? Well, if you look carefully we are doing the exact same thing, only doing away with writing the intermediate steps. As in the normal way, we have been faithful in our differentiation and integration. Whichever term we chose to differentiate/integrate at the start, we do it all the way. The alternating sign merely accounts for the “-“ sign in the formula presented in the 3rd equation. Try convincing yourself!

If you notice, you may realize that in some situations, we can go on differentiating/integrating forever – when do we stop? Here’s an example to illustrate this point:

Let’s choose to integrate x, since ln x is difficult to integrate. Haha.

Unlike when we integrate a xblah term, we do not get a lovely 0 to signify the end of our journey. So how? If you did attempt to prove to yourself that the shortcut is in essence the same thing as the formula, you should be able to know when you should stop differentiating/integrating. The answer is when the product of terms in the same row is easy to integrate. Then you integrate that product.

In this case, we can stop at the 2nd row:

In general, the formula can be presented as such:

* Differentiate the u term repeatedly.

* Integrate the term repeatedly.

* Starting from the 2nd row, fill in the alternating signs “+, -, +, -…” repeatedly

*At each row, check to see if the product can be easily integrated. If yes, stop there, and integrate that product. If no, continue down the columns.

*Multiply the terms as shown in the table above and factor in the relevant signs in the 3rd column. The arrow in blue is when the product of the terms in that row is easy to integrate. Which is also when we stop.

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