In calculus, we learn to integrate products like and using the technique integration by parts. We use the formula,
and using LIATE (Log-InvTrig-Algebraic-Trig-Exp), choose the right-leaning term as (e.g. ) and the left-leaning term as (e.g. ). We integrate and differentiate respectively to get and respectively, substitute into the formula, simplify the expression and carry on our calculations. Rather than confuse you with which expression to use as and , I propose a simpler presentation.
Before we discover this presentation, I want to clarify that this is just a simpler presentation of the same technique. The concept is the same. The presentation simply makes more intuitive sense. Also, I might derive the formula in another post. Integration by parts is essentially what I call the “reverse product rule”, and I might elaborate more next time. For now, lets take a look at the formula one more time, but this time I highlighted in green and and red and , and swapped the order of multiplying (doesn’t change the formula mathematically):
Notice that gets Integrated and remains the Same in the first term, . Notice that gets Integrated and gets Differentiated in the second term, . Thus, we get this idea of Integrate * Same – Integrate * Differentiate, or succinctly put,
As mentioned, this is essentially the same method, but using these letters makes it easier for us to choose which terms to use. Let’s integrate as an example.
Between and , the right-leaning term is . Thus, we Integrate that to get . The left-leaning term is . Thus, we Differentiate that to get .
Using IS-ID, we plug in the relevant letters to get
Integrating the remaining portion, we get our final result,
I hope this alternative presentation will help you more effectively and integrate by parts with better understanding. As an exercise to the reader, find . The final answer, for your reference, is and you get bonus marks if you can explain how this answer can be improved.