Integration by Parts - KNOW THIS FOR YOUR HSC!!!!!

'Integration by Parts' is a method for finding the integral of two functions multiplied together. If you're doing Extension 2 in your Higher School Certificate (HSC), and in fact most advanced high school math exams for senior year students, this is a crucial rule to know. It will undoubtedly show up in some part of HSC exam Question 1, which usually deals with integration.

The integration by parts rule -


I kinda like the first way of writing it. It's just easier to say in your head and have it stick so you can just have it up in your memory for your exam. The second way with f(x) and g(x) is good when you are first learning this method, perhaps, because it helps visualise two separate functions and what's happening with these functions. It's really up to you which one you like, but I will be using the 'u' and 'v' notation rather than f(x) and g(x).

So here's an example of how to use integration by parts, and then after this example from the 2009 HSC Extension 2 exam, I will then show you where this rule came from. It's nice to see how everything connects, but honestly that is a luxury you may not necessarily have time for if you're just trying to cram in as much stuff as possible before your exams, so I have put it at the end of this post just so you can see how everything comes together and relates.



Now this is where the integration by parts rule comes from...

The product rule for differentiation!

Well, the product rule for differentiation is another rule you will have to know for your exam. If you don't know it, learn it asap. If you do know it, whew! :)

So the rule for integration by parts comes simply from integrating the product rule. Yep, that's it!

So, really, if you are in your exam and forget what the rule for integration by parts is, you can find it by integrating the product rule. I wouldn't recommend this, though. Just memorise them both. Time in an exam can be much better spent.

So here are the two ways of writing the product rule depending on which notation you like.

So if you integrate the first part of the product rule, it's like the integral is undoing the d/dx, because it's like what's the integral of the derivative of u*v? The integration and differentiation undo each other.
Any questions? Issues? Disputes? Confused? Just a bit of a Nigel and need a friend? Comment below!

xj