Question on basic trig substitution with x = sin theta

[RoboNerd]

Say I have the integral of [ 1 / ( sqrt( 1 - x^2) ] * dx . Now I was told by many people in videos that I substitute x = sin theta, and this has me confused.

Wouldn't I need to substitute x = cos theta instead? as x = cos theta on the unit circle instead of sin theta?

Thanks in advance for your input! I am looking forward to hearing from you.

 

[Orodruin]

It really does not matter. The substitution in itself has nothing to do with x being the x coordinate of the unit circle, it is just the name of a new integration variable. You can call it whatever you want, x, y, t, or $\xi$, it is all the same.

 

[member 587159]

Say I have the integral of [ 1 / ( sqrt( 1 - x^2) ] * dx . Now I was told by many people in videos that I substitute x = sin theta, and this has me confused.

Wouldn't I need to substitute x = cos theta instead? as x = cos theta on the unit circle instead of sin theta?

Thanks in advance for your input! I am looking forward to hearing from you.

Orodruin's reply is a very good one. The only thing you use is the identity cos^2x + sin^2x = 1, so you can substitute both cos(t) or sin(t) for x.

 

[RoboNerd]

why is that even possible?

I was watching an MIT Opencourseware lecture and the professor explicitly said that the trig substitutions are tied with x=cos theta and y = sin theta on the unit circle.

The lecture is at:

Thanks for your input.

 

[RoboNerd]

Orodruin's reply is a very good one. The only thing you use is the identity cos^2x + sin^2x = 1, so you can substitute both cos(t) or sin(t) for x.

I understand if I substituted x = 1 - sin^2 x because that rests on the pythagorean identity, but that is obviously not the case.

 

[Orodruin]

why is that even possible?

It is just a substitution of variables, you can use any invertible smooth function for this. The only thing is finding a function so that the resulting integral is simpler. This has nothing to do with polar coordinates or parametrising the unit circle - it is just a variable substitution

 

[micromass]

Do you agree that

$$\int \frac{dx}{\sqrt{1-x^2}} = \int \frac{dy}{\sqrt{1-y^2}}$$

 

[RoboNerd]

we just say in this case that x = y, so yes?

 

[RoboNerd]

but why do they float this idea round? it has to be based off the theory that x = rcos theta

 

[Orodruin]

but why do they float this idea round? it has to be based off the theory that x = rcos theta

It has nothing to do with that. It is just a variable substitution.

 

[RoboNerd]

So how does this variable substitution work, then?

 

[Orodruin]

The same way as any variable substitution. If you have an integral
$$
\int_a^b f(x) dx
$$
and introduce a one-to-one function $x = g(t)$, then
$$
\int_a^b f(x) dx = \int_{g^{-1}(a)}^{g^{-1}(b)} f(g(t)) g'(t) dt.
$$
In your case, $f(x) = 1/\sqrt{1-x^2}$ and $g(t) = \sin(t)$.

 

[RoboNerd]

thank you for your help!