Question

Help with trig sub for `int_(xdx)/(4-x^2)`. Solution is resulting in a `-ln|costheta|+C` before final answer and I'm not seeing where it comes from?

So we have to solve this problem both without and with trig substitution. I did it without trig sub no problem and got the correct answer, but actually using trig sub is giving me issues because the book's simplification at the end is confusing me.

So, I know that I just need to treat the 4-x^2 as if it was a square root, and I've done that so I get the problem as far as breaking it down to `ln|sec(theta)|+c` but quickly realized that wasn't going to result in a match for my answer that I got without using trig sub, so after checking that I didn't do something wrong, I went to my solution manual and saw that the book is changing that to `-ln|cos(theta)|+c` and then getting the values from the triangle which does result in the same answer as I got without trig sub, but I'm not seeing how that change is taking place. I know `sectheta` is the same as `(1)/(costheta)` but how is it then becoming a -ln value? Especially when it's inside absolute value bars?

Answer 1

`ln|sectheta|=-ln|costheta|.` Your answer is fine in this regard.

Explanation:

There's often confusion over this, but `ln|sectheta|=-ln|costheta|`.

Recall that `alnx=ln(x^a).` The same applies for absolute values; `aln|x|=ln|x^a|`.

Thus, `-ln|costheta|=-1ln|costheta|=ln|(costheta)^-1|=ln|1/costheta|=ln|sectheta|`.

Your answer is the same as the book's answer. In fact, `ln|sectheta|` is cleaner than `-ln|costheta|.`

Let's run through the solution anyways, up to the point before the final answer, when we go in terms of `x:` :

`x=2sintheta, x^2=4sin^2thetad theta, dx=2costhetad theta`

So, we have
`int(4sinthetacostheta)/(4-4sin^2theta)d theta=int(sinthetacostheta)/(1-sin^2theta)d theta=int(sinthetacostheta)/(cos^2theta)d theta=inttanthetad theta=ln|sectheta|+C` OR `-ln|costheta|+C` -- both are equally valid. Same thing.