With this general case, notice how sqrt(x^2 - a^2) prop sqrt(sec^2theta - 1). So, we can use the following substitution:
x = asectheta
dx = asecthetatanthetad theta
sqrt(x^2 - a^2) = sqrt(a^2sec^2theta - a^2) = atantheta
Thus, we have:
= int 1/(cancel(atantheta))*cancel(a)secthetacancel(tantheta)d theta
= int secthetad theta
Then just a little trick:
= int sectheta((sectheta + tantheta)/(sectheta + tantheta))d theta
= int (sec^2theta + secthetatantheta)/(sectheta + tantheta)d theta
Now, let:
u = sectheta + tantheta
du = secthetatantheta + sec^2thetad theta
Therefore:
= int1/udu
= ln|u|
= ln|sectheta + tantheta|
Recall that:
sectheta = x/a
tantheta = sqrt(x^2 - a^2)/a
Thus we have:
= color(green)(ln|x/a + sqrt(x^2 - a^2)/a| + C)
...which is perfectly acceptable. But, you could simplify this more and be a little sneaky.
= ln|(1/a)[x + sqrt(x^2 - a^2)]| + C
= ln|x + sqrt(x^2 - a^2)| + ln|1/a| + C
but since a is a constant... it gets embedded into C.
= color(blue)(ln|x + sqrt(x^2 - a^2)| + C)
So if you see Wolfram Alpha give you this answer, that's why.
