Inverse Substitution: Solving for z in Terms of x

[jumbogala]

Homework Statement

Let's say you make the inverse substitution x = 2tan(z) in some integral.

Let's say you evaluate the integral get something like like 4sin(z). How do you put z back in terms of x?

Homework Equations

The Attempt at a Solution

I can do it by saying arctan(x/2) = z, but my teacher said we are not allowed to state the answer that way. I'm not sure how else you would do that...?

 

[Dick]

Draw a right triangle where tan(x)=x/2. For example, let the side opposite the angle z be x and the side adjacent be 2. Now use Pythagoras to find the hypotenuse. What then is sin(z) in terms of x? It's opposite over hypotenuse, right?

 

[jumbogala]

Ohh okay, so in this case you would end up with 4x / sqrt(x^2 + 4).

Thanks!

 

[Dick]

Exactly. 4*sin(arctan(x/2))=4x/sqrt(x^2+4). Drawing a triangle is nice way to derive stuff like that without memorizing it.