Trigonometric Substitution Definition

Question

Trigonometric substitution is defined as the method of replacing variables of integration with trigonometric functions. What I don’t understand is how is it that in the hypotenuse of a right triangle, $\sqrt{a^2+x^2}$, the value x can be defined as the product of $a\tan\theta$. If someone could shoot me an explanation that would be awesome.

Answer 1

First, a notation clarification: when you have $\int \sqrt{x^2+4} dx$ and you take $x=2\tan(\theta)$, $\theta$ is the new integration variable, so it isn't constant.

Moving on, there are two new things going on with trig substitution that don't really have anything inherently to do with trig. The first is that it is usually the first place where students see substitutions written as $x=f(u)$ rather than $u=f(x)$. You can do this in problems without trig, and conversely you do not have to do this when you do trig sub. For example in this problem you could have done $\theta=\arctan(x/2)$. But you can see why this is clunky; think about all the extra algebra that it takes to actually plug that into the integral.

The second is that you are even allowed to just bring trig into the mix when there's no trig function in the original problem. The gist here is that you can basically do whatever substitution you want. Technically you have to replace $x$ by a one-to-one function of the new integration variable, and in a definite integral the range of this function must include the domain of integration. But in practice you usually don't have to think about those requirements. (But just to show an example where you do, figure out why you cannot do $x=\sqrt{u}$ on $\int_{-1}^1 x^2 dx$...figure out which of those rules got broken.)