Evaluate the following integral:

Question:

Evaluate the following integral:

$\int \frac{3 x+5}{x^{3}-x^{2}-x+1} d x$

Solution:

$I=\int \frac{3 x+5}{x^{3}-x^{2}-x+1} d x=\int \frac{3 x+5}{(x-1)^{2}(x+1)}$

$\frac{3 x+5}{(x-1)^{2}(x+1)}=\frac{A}{x-1}+\frac{B}{(x-1)^{2}}+\frac{C}{x+1}$

$3 x+5=A(x-1)(x+1)+B(x+1)+C(x-1)^{2}$

Put $x=1$

$-3+5=4 C$

$2=4 \mathrm{C}$

$C=\frac{1}{2}$

Put $x=0$

$5=-A+B+C$

$A=\frac{1}{2}$

$\int \frac{3 x+5}{(x-1)^{2}(x+1)} d x=\frac{1}{2} \int \frac{d x}{x-1}+4 \int \frac{d x}{(x-1)^{2}}+\frac{1}{2} \int \frac{d x}{x+1}$

$=-\frac{1}{2} \ln |x-1|-\frac{4}{(x-1)}+\frac{1}{2} \ln |x+1|+C$

$=\frac{1}{2} \ln \left|\frac{x+1}{x-1}\right|-\frac{4}{(x-1)}+C$

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