Integration By Parts

Integration By Parts

JC Mathematics

Today, I’ll share a little something about Integration by Parts. I want to share this because I observe that several students are over-reliant on the LIATE to perform Integration by Parts. This caused them to overlook/ appreciate its use.

Lets start by reviewing the formula for Integration by Parts

\int v \frac{du}{dx} ~dx = uv - \int u \frac{dv}{dx} ~dx

I like to share with students that Integration by Parts have two interesting facts.

  1. It allows us to integrate expression that we cannot integrate. Eg. \text{ln}x or inverse trigonometric functions. This is also closely related to how LIATE is established.

  2. This is closely related to point 1. That is, instead of trying to integrate the expression, we are differentiating it instead. And that itself, is a very important aspect.

Next, I like to point out to students that LIATE is a general rule of thumb.
GENERAL – because it does not work 100% of the times. And today, I’ll use an example to illustrate how LIATE actually fails.

\int \frac{te^t}{(t+1)^2} ~ dt

Here we observe two terms \frac{t}{(t+1)^2} and e^t. Going by LIATE, we should be differentiating \frac{t}{(t+1)^2} and integrating e^t.

\int t^2 e^{t} ~ dt
= \frac{t}{(t+1)^2} e^{t} - \int \frac{(1)(t+1)^2 - t(2)(t+1)}{(t+1)^4} e^t ~dt

Now observe what happened here, after applying integration by parts, it got “worst”. We are stuck to integrating \frac{(1)(t+1)^2 - t(2)(t+1)}{(t+1)^4} with an exponential… This should raise some question marks. But we did follow LIATE.

I’ll leave students to try out this on their own. And feel free to ask questions here. Have fun!

Random Questions from 2016 Prelims #7

Random Questions from 2016 Prelims #7

JC Mathematics

VJC P1 Q9

(i) Sketch the graph with equation x^2 +(y-r)^2 = r^2, where r >0 and y \le r

A hemispherical bowl of fixed radius r cm is filled with water. Water drains out from a hole at the bottom of the bowl at a constant rate. When the depth of water if h cm (where h \le r).

(ii) Use your graph in (i) to show that the volume of water in the bowl is given by V = \frac{\pi h^2}{3} (3r-h).

(iii) Find the rate of decrease of the depth of water in the bowl, given that a full bowl of water would become empty in 24 s,

(iv) without any differentiation, determine the slowest rate at which the depth of water is decreasing.