Tricks. More tricks.
As we have tried to tell you again and again, integration involves TRICKS. Hopefully, you would have picked up several tricks and developed several tricks of your own by now. this section is all about tricks in integration.
Immensely helpful for testpreppers. Do not miss.
Immensely helpful for testpreppers. Do not miss.
It is like magic, in a sense.
Page Walkthrough
Turning the Big Gun AroundIn the last few sections, we have learned how to evaluate an indefinite integral. And the evaluation of definite integrals follows. We have learned how to use trigonometry, algebra, and anything we have to throw at that stupid elongated ‘S’.
We have learned several things—differentiation formulas can sometimes be turned around to attack integration problems. Here, we learn how to use the Product Rule to solve a broader range of integration problems. If you still remember, the Product Rule states—if h(x)=f(x)*g(x), then h’(x)=f’(x)g(x)+f(x)g’(x). Somebody realized this could be a perfect weapon for integration. Think about this: integrate xsin(x). We feel that this might be one of the two terms on the right hand side of a Product Rule application. Suppose sin(x)=f’(x) and x=g(x). Now, f’(x)g(x)=h’(x)f(x)g’(x). And now we integrate. The integral of the left hand side (what we are trying to find) must be equal to the integral of the right hand side, which is h(x) minus the integral of f(x)g’(x) plus some constant. So, we have transferred the problem into finding the integral of f(x)g’(x). Now you are asking: so what? We have traded one problem for another! That’s where the tricks come in. When you apply this procedure, oh, it’s called integration by parts, you need to make sure that evaluating the integral of f(x)g’(x) is easier than the original problem. Up there, we set sin(x)=f’(x) and x=g(x). Now, f(x)g’(x) is just –cos(x). You should be able to know the integral immediately. Down below is a table that will help:
After trying out for the best f(x) and g'(x), bam! Solve the puzzle. And take a break.

List of TricksTrick #1: If the function we are trying to integrate is “x” times “something”, (e.g. xsin(x)), it is usually a good idea to let x=g(x), and the rest to be f’(x). This is because the derivative of x is 1. The problem is reduced to finding the integral of that “something”.
Trick #2: If there is an e^x in the equation, it can serve both as an f’(x) or g(x). This is because both the integral and the derivative of e^x is itself. We can decide if we want to have the derivative of the rest of the function or its integral. Usually, we want its derivative, as it is often much easier to find. Trick #3 (not technically a trick): If there is a square root in the equation, integration by parts would often not work. Trick #4: Sometimes, we can apply integration by parts many, many times. Remember that we have transformed the function to be integrated into another one? We can transform it again, and again. Until we find the result. But take a note—if you can’t find the answer in three tries of integration by parts, you’d better try the other way. Now have a cup of coffee and congratulate yourself on what you have just learned!
