Mechanical PE Exam Topic Concepts

Integration by Parts – The Fast and Easy Way!

integration by parts

Work smarter, not harder

~Scrooge McDuck

Integration by Parts

In this blog you will learn how to do integration by parts the fast and easy way!  First of all… the majority of problems on the PE exam will not require calculus, but it is possible.  Because it may be required… and because success on the PE exam is all about working problems quickly… I wanted to share this VERY SIMPLE and VERY FAST method for doing integration by parts!  In fact, it is so simple that you will wonder why this method is not commonly taught in calculus courses (I don’t know the reason either).  So even if you don’t need this method on the PE exam it is a great tool to have… and you can use it to impress your friends!


Why would I care… I don’t solve calculus problems by hand!

So you may be thinking… if any integration appeared on the PE exam I would look up the solution or use my calculator to solve the integral.  Though I do recommend you have a good math reference book for the exam (I highly recommend this one as a good overall math reference), I honestly believe that you could solve some basic integration by parts problems faster with the method I am about to show you because it takes a while to find the solution in the book.  As far as using the calculator, you must remember that the NCEES has a limited list of approved calculators.  So the calculator may not have all the functions for calculus.

What is integration by parts?

Before I get into the details of this VERY SIMPLE and VERY FAST method for doing integration by parts, I want to briefly remind you about the basic idea of integration by parts.  Integration by parts is based on the concept of the product rule of differentiation.  Integration by parts is a method commonly used when the function to be integrated is a product of an algebraic function and a trigonometric or exponential function.

Just tell me about this VERY SIMPLE and VERY FAST method!

OK… so let’s just get to the point!  What is this VERY SIMPLE and VERY FAST method?  The method presented here is one that I call the tabular method, because a simple table is developed to quickly develop the solution to the integral.  One nice advantage is that the solution is developed in one step regardless of the problem (no more repeated integration by parts in the same problem… which typically happens).

I am going to explain the process with an example.

int by parts example 1

I will illustrate the process by solving the integral shown.  The function is a product of an algebraic function and an exponential function, so integration by parts applies.

Step 1: Make the table

integration by parts example 2

The first step is to construct a very simple table.  The first column is the term in the integral labeled u and the second column is the term labeled dv.  Differentiate the ‘u’ term in the first column and continue until the derivate becomes zero.  Integrate the ‘dv’ term and continue to the end of the first column.

Step 2: Cross-multiply and sum

int by parts example 3

The second step, which is the last step, is to cross-multiply and sum the results.  Before you do that you need to alternate positive and negative signs (+ and – as shown)… those will be used in the summation step.  Now simply multiply the terms as shown with the red arrows and complete the sum of all the terms (don’t forget the + and – signs).  Add the integration constant C and you are done!  Very fast, very simple, and all completed in one step!

Action Steps

What should you do now?  Go grab a calculus book and look for some example problems on integration by parts.  Solve those problems using this method to prove to yourself how much faster this process is compared to the standard method.

Let me know your thoughts! 

I hope this helped you improve your efficiency in integration!  I know it may be a little confusing from this blog post alone, so if you have questions or comments please let me know.  I can always add a couple more examples to help clarify if needed.  Also, please take a moment to share this post with others!  You can click any of the ‘share the knowledge’ tabs.  That helps me a lot!

As always feel free to leave a comment!  Be sure to check out the Mechanical PE Academy Facebook page for updates and exclusive content!

9 comments on “Integration by Parts – The Fast and Easy Way!

  1. This is so beautiful. Usually students take a lot of time to comprehend the repeated integration procedure, let alone to execute it!

    This method is going to enhance my teaching.


    1. Thanks, Geoffrey! I am not sure why it is not taught. The repeated integration typically required is very complicated! I am glad this helped!

  2. This is a standard method taught in all high school calculus courses as a shortcut to iterated integration by parts. Virtually any calculus book in existence should contain this method. It’s useful, but you didn’t invent anything unique.

  3. It is really a nice and helpful piece of information. I’m glad that you shared this helpful info with us. Please keep us informed like this. Thanks for sharing.

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