Question

How do I find the antiderivative of `f(x) = (5x)/(10(x^2-1))`?

Answer 1

I would first manipulate the argument to get it in a form which is easier to integrate:
Simplifying the `5` and `10` and transforming `x^2-1` in a product you get:

`int(5x)/(10(x^2-1))dx=int(x)/(2(x-1)(x+1))dx=`

I then rearrange to get a sum introducing an additional `1/2`;

`=int1/4*(1/(x-1)+1/(x+1))dx=`

(which is equivalente to the starting one: `int(x)/(2(x-1)(x+1))dx`)

And finally:

`=int1/4*(1/(x-1)+1/(x+1))dx=1/4[ln(x-1)+ln(x+1)]+c`
or
`=1/4[ln(x^2-1)]+c`