Question

How do you integrate `int 1/(xsqrt(x^2-1) )dx` using trigonometric substitution?

Answer 1

`int \ 1/(xsqrt(x^2-1)) \ dx = arcsecx + C`

Explanation:

We seek:

`I = int \ 1/(xsqrt(x^2-1)) \ dx`

Let us attempt a substitution of the form:

`sectheta=x`

Then differentiating wrt `x` we have:

`sectheta tan theta (d theta)/dx = 1`

Substituting into the integral we have:

`I = int \ 1/(sec theta sqrt(sec^2theta-1)) \ sectheta tan theta \ d theta`
`\ \ = int \ 1/(sec theta sqrt(tan^2theta)) \ sectheta tan theta \ d theta`
`\ \ = int \ 1/(sec theta tan theta) \ sectheta tan theta \ d theta`
`\ \ = int \ d theta`
`\ \ = theta + C`

Restoring the substitution gives:

`I = arcsecx + C`