Question

How do you integrate `1/(x^2sqrt(x^2 - 4))` using trig substitution?

I believe the correct substitution is 2sec`theta` but I can't get the equation simple enough to try u-substitution or just integrate in trig.

My simplest equation:
`int(2secthetatantheta)/(4sec^2thetasqrt(4tan^2theta))d theta`

Answer 1

`sqrt(x^2-4)/(4x)+C.`

Explanation:

I will proceed further from where the Questioner has derived the

simplest integral, i.e.,

`I=int(2secthetatantheta)/{4sec^2theta*2tantheta)d(theta)`

`=1/4int1/sectheta d(theta)`

`=1/4intcosthetad(theta)`

`=1/4sintheta`

Now, `sectheta=x/2 rArr costheta=2/x`

`:. sintheta=sqrt(1-cos^2theta)=sqrt(1-4/x^2)=sqrt(x^2-4)/x`

`:. I=sqrt(x^2-4)/(4x)+C.`

Enjoy Maths.!