Integrate the rational function x^ 3 +x+1 x^ 2 -1 - Vidyadip

Question

Integrate the rational function $\frac{x^{3}+x+1}{x^{2}-1}$

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Answer

Given function is $\frac{x^{3}+x+1}{x^{2}-1}$
Dividing $(x^3 + x + 1)$ by $x^2 - 1$, we get,
$\frac{x^{3}+x+1}{x^{2}-1}=x+\frac{2 x+1}{x^{2}-1}$
Let $\frac{2 x+1}{x^{2}-1}=\frac{A}{(x+1)}+\frac{B}{(x-1)}$
Now, $2x + 1 = A(x – 1) + B(x + 1)$ ...(i)
Substituting x = 1 and -1 in equation (i), we get,
$A=\frac{1}{2}$ and $B=\frac{3}{2}$
Thus, $\frac{x^{3}+x+1}{x^{2}-1}=x+\frac{1}{2(x+1)}+\frac{3}{2(x-1)}$
$\Rightarrow~~\int \frac{x^{3}+x+1}{x^{2}-1}=\int x d x+\frac{1}{2} \int \frac{1}{(x+1)} d x+\frac{3}{2} \int \frac{1}{(x-1)} d x$
= $\frac{x^{2}}{2}+\frac{1}{2} \log |x+1|+\frac{3}{2} \log |x-1|+C$

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