Question

Please help me!

`\int\limits { (1)/( x√(x^2-1) ) } \, dx`

Answer 1

Substitute
`z=√(x^2-1)`, so that
`x^2=z^2+1`. Then
`\mathrm dz=\frac x{√(x^2-1)}\,\mathrm dx`. You have


`\displaystyle\int(\mathrm dx)/(x√(x^2-1))=\int\frac x{x^2√(x^2-1)}\,\mathrm dx=\int(\mathrm dz)/(z^2+1)`

This is a standard integral, and the antiderivative is


`\arctan z+C`

(you can verify with a trig sub of
`z=\tan u`, for instance). Transforming back to a function of
`x`, you get


`\displaystyle\int(\mathrm dx)/(x√(x^2-1))=\arctan(√(x^2-1))+C`