Trig functions within Trig functions

[Cajstyle]

Homework Statement

I'm given the problem: $$\int$$$$\frac{\sqrt{y^{2}-25}}{y}$$ dy

Homework Equations

I set y = 5sec(u), and solve for the subsitution.

The Attempt at a Solution

At the culmination of my solution, I achieve: 5tan(u) - 5u + C

Here is my dilemma, u stands for arcsec($$\frac{y}{5}$$) -- How do I deal with tan(arcsec(..)), or any "trig within trig" setup? I've gone all this time without actually ever figuring out how to solve for these.

 

[Fightfish]

You can use a right-angled triangle to aid you in visualizing what is going on. For instance, arcsec(y/5) returns an angle in a right-angled triangle where the adjacent side is of length 5 and the hypotenuse is of length y. Then, you can easily calculate the tangent of that angle.

 

[Cajstyle]

I was previously using this as a method to solve them but I ran into a few instances of having different answers than the book. I'll look further into it, thank you for the response.

 

[hunt_mat]

Use
$$1+\tan^{2}x=\sec^{2}x$$
To comvert the tan into a sec.