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

1 Answer
Jan 29, 2015

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#