How does trigonometric substitution work?

I have read my book, watched the MIT lecture and read Paul's Online Notes (which was pretty much worthless, no explanations just examples) and I have no idea what is going on with this at all.

I understand that if I need to find something like $$\int \frac { \sqrt{9-x^2}}{x^2}dx$$

I can't use any other method except this one. What I do not get is pretty much everything else.

It is hard to visualize the bounds of the substitution that will keep it positive but I think that is something I can just memorize from a table.

So this is similar to u substitution except that I am not using a single variable but expressing x in the form of a trig function. How does this not change the value of the problem? To me it seems like it would, algebraically how is something like $$\int \frac { \sqrt{9-x^2}}{x^2}dx$$ the same as $$\int \frac {3\cos x}{9\sin^2 x}3\cos x \, dx$$

It feels like if I were to put in numbers for $x$ that it would be a different answer.

Anyways just assuming that works I really do not understand at all what happens next.

"Returning" to the original variable to me should just mean plugging back in what you had from before the substitution but for whatever unknown and unexplained reason this is not true. Even though on problems before I could just plug back in my substitution of $u = 2x$, $\sin2u = \sin4x$ that would work fine but for whatever reason no longer works.

I am not expected to do some pretty complex trigonometric manipulation with the use of a triangle which I do not follow at all, luckily though this process is not explained at all in my book so I think I am just suppose to memorize it.

Then when it gets time for the answer there is no explanation at all but out of nowhere inverse sin comes in for some reason.

$$\frac {- \sqrt{9-x^2}}{x} - \sin^{-1} (x/3) +c$$

I have no idea happened but neither does the author apparently since there is no explanation.

Solution 1:

It might help to not write $x$ for the before and after. That's misleading. What you're doing is writing $x=3\sin(\theta)$. You're inventing a new variable $\theta$ that relates to $x$ in a way that lets you use trig identities to solve your problem. Plugging in values after your transformation shouldn't give you the same values, since the new integral is in a different variable. We have: $$x=3\sin(\theta)$$ $x$ is a function of $\theta$. Differentiating we get: $$\frac{dx}{d\theta}=3\cos(\theta) \implies dx=3\cos(\theta) d\theta$$

So rewriting the integral in terms of our new variable, we get

$$\int \frac {\sqrt{9-9\sin^2(\theta)}}{9\sin^2(\theta)}\cdot3\cos(\theta)d\theta$$

Now we can use trig identities to evaluate this new integral and get rid of the troublesome square root, which is the reason we bothered with this substitution in the first place.

$$\int \frac {\sqrt{9\cos^2(\theta)}}{9\sin^2(\theta)}\cdot3\cos(\theta)d\theta$$ $$\int \frac {\cos^2(\theta)}{\sin^2(\theta)}d\theta$$ expanding the numerator as $1-\sin^2(\theta)$: $$\int \csc^2(\theta)-1 d\theta$$ evaluating, we get: $$-\cot(\theta)-\theta+C$$

But our integral was in terms of $x$. While this substitution was helpful, we're only halfway done, since we don't know what the expression looks like in our original variable. So now we need to sub in our original variable. $x=3\sin(\theta)$, so to get things in terms of $x$ we'd like to get as many $\sin(\theta)$s as we can. $\cot(\theta)=\frac{\cos(\theta)}{\sin(\theta)}=\frac{\sqrt{(1-\sin^2(\theta)}}{\sin(\theta)}$. By our substitution, $\sin(\theta)=x/3$. So we get $$-\frac{\sqrt{1-\frac {x^2} 9}}{\frac x 3}-\theta+C$$ We still have a $\theta$ we need to get rid of. To do this, we can just reverse our substitution. That is, $x=3\sin(\theta) \implies \theta=\sin^{-1}(x/3)$. C is still an arbitrary constant and so can stay. After simplifying the fraction, we get: $$-\frac{\sqrt{9-x^2}}{x}-\sin^{-1}(x/3)+C$$ Which is the answer. Hopefully this clears some things up.

Solution 2:

The following theorem is useful to keep in mind when making substitutions:

If function $x=\phi\left(t\right)$ satisfies the following conditions:

1) $\phi\left(t\right)$ is a continuous one-to-one function defined on the interval $\left[\alpha,\beta\right]$ and having a continuous derivative there.

2) values of $\phi\left(t\right)$ are contained within the interval $\left[a,b\right]$

3) $\phi\left(\alpha\right)=a$ and $\phi\left(\beta\right)=b$

Theorem. then for every function $f\left(x\right)$ continuous on $\left[a,b\right]$ the following formula for the change of variable in a definite integral holds:$$\int_{a}^{b}f\left(x\right)dx=\int_{\alpha}^{\beta}f\left[\phi\left(t\right)\right]\phi'\left(t\right)dt$$

Solution 3:

You must use a name other than $x$ when you substitute, maybe say let $x=3\sin t$. (It makes no sense to let $x=3\sin x$.) Using $x$ for the substitution is probably one of the big reasons for being confused about the process.

We have then $dx=3\cos t\,dt$, and $\sqrt{9-x^2}=\sqrt{9(1-\sin^2 t}=3\cos t$. So we arrive at $$\int \frac{\cos^2 t}{\sin^2 t}\,dt.$$ To evaluate, use $\cos^2 t=1-\sin^2 t$. So we want $\int \frac{1-\sin^2 t }{\sin^2 t}\,dt$.

This simplifies to $\int \left(\frac{1}{\sin^2 t} -1\right)\,dt$, and then to $\int(\csc^2 t-1)\,dt$.

Now it is useful to remember that $-\cot t$ is an antiderivative of $\csc^2 t$. We conclude that our integral is equal to $$-\cot t + t+C.$$ Finally, we want to go back to $x$. Remember that $x=3\sin t$, so $\sin t=x/3$, and therefore $t=\sin^{-1}(x/3)$.

To express $\cot t$ in terms of $x$, use the fact that $\frac{\cos t}{\sin t}$. We have $\sin t=x/3$, and $\cos t=\sqrt{1-(x/3)^2}=\frac{1}{3}\sqrt{9-x^2}$. When we put the pieces together we get the expression that you quoted.