Am I understanding trigonometric solution correctly?

[1MileCrash]

More of a general inquiry..

I was given some homework to do on trigonometric substitution. It looks to me like the goal is always to get an integral of 1 by itself, then replacing theta that results from integrating with it's x equivalent?

On my homework, as long as I made the right choice for substituting, I always got the integral of cos/cos or similar, which was 1, integrated was theta, which was found by solving my substitution for theta.

 

[cragar]

The point of a trig substitution is to get rid of square roots or radicals. Or something squared plus something. In your case it became one but that doesn't happen all the time.

 

[1MileCrash]

Okay, I've been working with a few more and I often get results like:

sin(arcsin t)
cos(arcsin t)

And results like that. Is there a good way to remember the algebraic equivalent for those?

 

[Mark44]

Okay, I've been working with a few more and I often get results like:

sin(arcsin t)
cos(arcsin t)

And results like that. Is there a good way to remember the algebraic equivalent for those?

For the first, sin(arcsin t) = t, subject to possible restrictions of the domain of the sine function.

For the second one, I personally don't think it's worthwhile to clutter my brain with a formula. Instead, sketch a right triangle one of whose acute angles represents arcsin(t). So you have a right triangle with $\theta$ as one of the acute angles.

Since $\theta$ = arcsin(t), then t = sin($\theta$). You can label the opposite side as t, and the hypotenuse as 1. What's the length of the other side (the adjacent side)? IOW, what is cos($\theta$)? That will be the same as cos(arcsin(t)).

 

[1MileCrash]

Square root one minus x squared?

 

[Mark44]

Square root one minus x squared?

You were asking about cos(arcsin(t)). There wasn't an x anywhere in sight.