How to know when you DON'T need to use substitution (integrals)

• Feb 16th 2011, 09:31 AM
Marconis
How to know when you DON'T need to use substitution (integrals)
We've been doing substitution for a few weeks now, and it's so embedded in my mind that when I look to take an integral, I immediately look for a substitution. I was just trying to do a simple problem and was looking at it for several minutes until I finally realized a substitution was completely unnecessary.

This may seem like a silly question, but what kinds of things separate having to use a u-substitution to not having to use it? What should I be looking for as quick indicators?
• Feb 16th 2011, 09:45 AM
TheCoffeeMachine
Quote:

How to know when you DON'T need to use substitution (integrals)
When you can do the integral without it, I think! If I look at an integral and cannot see how it can be done without making one or another substitution, I proceed to make one.
• Feb 16th 2011, 09:47 AM
Marconis
Ha, I guess that's easy enough. Then I start to doubt myself and think I am making an incorrect sub. or something. Guess I should just go with my gut then take the derivative to see if I am correct.
• Feb 16th 2011, 10:19 AM
TheCoffeeMachine
Quote:

Originally Posted by Marconis
Then I start to doubt myself and think I am making an incorrect sub.

I'm not good at these things myself, but I think toying with different substitutions eliminates this problem.
Say, if you have $\displaystyle \int \sin^2{x}\cos^2{x}\;{dx}$, write it as $\displaystyle \int \sin^2{x}\left(1-\sin^2{x}\right)\;{dx}$ and sub $\displaystyle u = \sin{x}$; or
write it as $\displaystyle \int \cos^2{x}\left(1-\cos^2{x}\right)\;{dx}$ and let $\displaystyle u = \cos{x}$; or write it as $\displaystyle \frac{1}{8}\int(1-\cos{4x})\;{dx}$ to see
that no sub is needed. This way you'll learn when a sub is necessary and/or is more efficient than another.
• Feb 16th 2011, 10:30 AM
Marconis
That makes more sense. Thank you!
• Feb 16th 2011, 01:46 PM
DrSteve
There are no absolute rules for when a substitution will work. There are some tricky integrals out there where a clever substitution that seemingly comes out of thin air can be used to evaluate the integral.

That said, if you understand that performing a substitution is doing the chain rule backwards, then usually a substitution is used when you have a composition of 2 functions, and the derivative of the inner function is also inside the integrand. If you don't have this situation, then you probably want to try other methods before trying to find a clever substitution.

One more note: If you were to form an integral at random, then it is extremely unlikely that substitution would work.
• Feb 16th 2011, 01:59 PM
Marconis
Quote:

Originally Posted by DrSteve

That said, if you understand that performing a substitution is doing the chain rule backwards, then usually a substitution is used when you have a composition of 2 functions, and the derivative of the inner function is also inside the integrand. If you don't have this situation, then you probably want to try other methods before trying to find a clever substitution.

This bit right here makes it easier to understand.

Thanks Dr. Steve...good to see another NYC member on here!
• Feb 16th 2011, 02:07 PM
DrSteve
Quote:

Originally Posted by Marconis
This bit right here makes it easier to understand.

Thanks Dr. Steve...good to see another NYC member on here!

One thing that you may find helpful is to write down a bunch of simple functions like $\displaystyle x, \ln x, e^x, \sin x,$ etc. compose them, and then take the derivative of the inner function. You'll get a clear picture of when substitutions work.

Here's one example:

$\displaystyle f(x) =x^2, g(x)=\ln x, g'(x)=\frac{1}{x}, f(g(x))=(\ln x)^2$.

We get $\displaystyle \frac{(\ln x)^2}{x}$. By design this is a function which can be integrated by substitution.

NYC's the place to be, except maybe in the winter..
• Feb 16th 2011, 02:39 PM
Marconis
Thanks again. Wonderful response.
• Feb 16th 2011, 04:36 PM
Kels14
If you have a "plain old x" the term my calc teacher uses you don't have to use substitution such as the integral of sinx or 1/x but when you have something like the intergral of cos3x or e^6x you may need to use substitution.