<s>[INST] Integrate $4x/(x^4-1)$ dx I'm having trouble solving the integral 
$$\int_{5/4}^{13/12}\frac{4x}{x^4-1}\,dx$$
I have a feeling it is to do with the log integration identity but can't seem to manipulate it without involving complex numbers, something outside of this module (so I should be able to solve without them).
Any help appreciated, thanks in advance.
 [/INST] $$\int \frac{4x}{x^4-1}dx$$
Partial Fraction Decomposition on $\frac{4x}{x^4-1} = \frac{4x}{(x^2+1)(x^2-1)}$ gives $\frac{a}{x^2+1} + \frac{b}{x^2-1} = 4x \implies a(x^2-1)+b(x^2+1) = 4x$. Solving this gives $a=-2x, b=2x$
$$=\int\frac{2x}{x^2-1}-\int\frac{2 x}{x^2+1}dx$$
$$=\log|x^2-1| -\log(x^2+1) + C$$
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