Why trig substitution works

In other words, we would need to use the substitution that we did in the problem. Notice as well that we could have used cosecant in the first case, cosine in the second case and cotangent in the third case.

Simply because of the differential work. Note that the root is not required in order to use a trig substitution. Instead, the trig substitution gave us a really nice of eliminating the root from the problem. In this section we will always be having roots in the problems, and in fact our summaries above all assumed roots, roots are not actually required in order use a trig substitution. We will be seeing an example or two of trig substitutions in integrals that do not have roots in the Integrals Involving Quadratics section.

Now, we have a couple of final examples to work in this section. Not all trig substitutions will just jump right out at us. Sometimes we need to do a little work on the integrand first to get it into the correct form and that is the point of the remaining examples.

Note however that if we complete the square on the quadratic we can make it look somewhat like the above integrals. Here is the completing the square for this problem. Note we could drop the absolute value bars since we are doing an indefinite integral.

Here is the integral. However it is. To see this we first need to notice that,. We do need to be a little careful with the differential work however. Here is the substitution work. This is now a fairly obvious trig substitution hopefully.

Here is that work. Because we are doing an indefinite integral we can assume secant is positive and drop the absolute value bars. Applying this substitution to the integral gives,. We can then compute the differential. Doing this gives,. Again, we can drop the absolute value bars because we are doing an indefinite integral. The integral then becomes,. So, the same integral with less work. However, it does require that you be able to combine the two substitutions in to a single substitution.

The single substitution method was given only to show you that it can be done so that those that are really comfortable with both kinds of substitutions can do the work a little quicker. Notes Quick Nav Download.

You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Example 1 Evaluate the following integral. Note, however, the presence of the absolute value bars.

These are important. Example 2 Evaluate the following integral. Example 3 Evaluate the following integral. Example 4 Evaluate the following integral.

However, the following substitution and differential will work. Example 5 Evaluate the following integral. Example 6 Evaluate the following integral. Example 7 Evaluate the following integral. By putting that in our new equation we have our new equation as. Seem familiar? We have seen that the only two things necessary to do a substitution is that. In the equation referenced you have a range between -3 and 3, as does 3 cos theta.

Obviously, they use different variables, so you are good. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. How does trigonometric substitution work? Ask Question. Asked 9 years, 5 months ago. Active 2 years, 10 months ago.

Viewed 7k times. What I do not get is pretty much everything else. Anyways just assuming that works I really do not understand at all what happens next. Michael Hardy 1. Add a comment. Active Oldest Votes. Robert Mastragostino Robert Mastragostino And yes I did use a memorized formula i.

You can figure it out properly, but it's a big enough problem on its own that I figured doing it out would sidetrack too far from the actual question. How is it not changing the value of everything? More importantly though how the heck does the arcsin stuff work? I don't see what happened.

I don't even really know how arc sin works. But afterwards I'm changing it back , so it doesn't matter. If I want to find the surface area of a cylinder, I solve the equivalent problem of the combined area of two circles and a rectangle. This is legal, if I can still make the cylinder out of those shapes afterwards. So I am changing how I represent the problem in terms of a different variable , but because I can undo that transformation everything works out.

Relative to the space we work in, it's the same function, same question. We change it back as whilst it's easier to solve we want an answer in real space, not our imagined space. Show 2 more comments.

Valentin Valentin 4, 17 17 silver badges 21 21 bronze badges. I don't know why phi of t means. Is that a function? Like f t? So the first thing I need to do is verify that the function is one to one. I do not know of any easy way to do this, maybe just make sure it isn't an absolute value or something like that? Then I need to make sure that the values of the function are in te interval. I don't really know what that means either, so I have to test each point or the end points? Then 3 is really weird and I have no idea at all what that means.

I see how using 3cost is wanted because then when you square it you can factor out a 9 of the radicand and get the form of a trig identity.

I am trying to find the integral of something so why am I only manipulating one side? To me it seems like I am just changing the problem. That's usually a good idea anyway. Therefore you are just exploiting the properties of trig functions to simplify the integrand before you substitute back at the end.

This is all just a different way of looking at the same number. Completely pointless but bear with me. We have seen that the only two things necessary to do a substitution is that a. Dennis Gulko Dennis Gulko 15k 1 1 gold badge 31 31 silver badges 55 55 bronze badges. I don't quite understand arcsin and such. DonAntonio DonAntonio k 17 17 gold badges silver badges bronze badges.

Sign up or log in Sign up using Google. Sign up using Facebook.