In order to solve integrals in calculus, you need a variety of tools at your disposal. One such tool is integration by parts, and is invaluable in situations where u-substitution does not work. As this is just one of a variety of my calculus tutorials, I highly recommend reading through some of them. For you to get the fullest understanding of how to integrate a calculus function through integration by parts, please read my tutorial on integration rules. Also, check to see that u-substitution won't better serve your needs; in most cases, it is the best course of action.
What is integration by parts?
Integration by parts is a calculus formula for solving integrals, but its formula is remarkably more complex to look at than that of u-substitution:
The formula comes from the product rule for derivatives, and how it is proven is shown below:
The method by which you begin using the calculus formula of integration by parts is similar to u-substitution: you must select a u. However, u doesn't have to have its derivative in the remainder of the integrand. Rather, you will make the rest of the integrand dv.
In order to choose the best u for the calculus formula for integration by parts, I recommend using the mnemonic device LIPET. Logarithms are the best choice, followed by inverse trigonometric functions, then polynomials, then exponential functions, and lastly trigonometric functions. This is due to the fact that logarithms and inverse trigonometric functions are difficult to integrate, and thus would serve better as u since you will be deriving it.
Let us say you wish to integrate the following calculus function using integration by parts:
By following LIPET, you would select u as x, the polynomial, and the dv as sin(x) dx, the trigonometric function. You would then find v and du:
Then, simply follow the calculus formula for integration by parts:
Be sure not to forget to add C to the end of the simplification! Also, keep track of the sign. As you come across more complex problems that require integration by parts, don't be afraid to use plenty of parentheses and make notes of your steps to reduce error.
I hope you now have a firmer understanding of the calculus formula for integration by parts, not only for how it is executed, but also from where it originated. It is truly a useful skill to master, so be sure you are comfortable with it as you approach a test on integration. Good luck with your studies!