Question

Solve : $\int \frac{2x}{(x^2+1)(x^2+3)}$

Answer 1

$\int \frac{2x}{(x^2+1)(x^2+3)}dx$

$=2\int \frac{x}{(x^2+1)(x^2+3)}dx$

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

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$\int \frac{x}{2(x^2+1)}$

substitute $u=x^2+1\to du=2xdx$

$=\frac{1}{2}\int \frac{1}{2u}du$

$=\frac{1}{4}\ln |x^2+1|$

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$\int \frac{x}{2(x^2+3)}$

substitute $u=x^2+3\to du=2xdx$

$=\frac{1}{2}\int \frac{1}{2u}du$

$=\frac{1}{4}\ln |x^2+3|$

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$=2(\frac{1}{4}\ln |x^2+1|-\frac{1}{4}\ln |x^2+3|)+C$