Question

Evaluate the integral with hyperbolic substitution. ?

Answer 1

`int1/(xsqrt(4-x^2))dx=-1/2lnabs((2+sqrt(4-x^2))/x)+"c"`

Explanation:

We want to find `1/(xsqrt(4-x^2))dx`

First substitute `x=2sinu` and `dx=2cosudu`.

Then

`int1/(xsqrt(4-x^2))dx=int1/((2sinu)2sqrt(1-sin^2u)) 2cosudu=1/2int(cscu) (1/(2cosu))(2cosu)du=1/2intcscudu`

This is a well known integral, so we get

`1/2intcscudu=-1/2lnabs(cscu+cotu)`

We know substitute back `u=arcsin(x/2)`

`-1/2lnabs(cscarcsin(x/2)+cotarcsin(x/2))`

This can be shown to be equal to

`-1/2lnabs((2+sqrt(4-x^2))/x)+"c"`