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Integrate the rational functions(2x)/((...

Integrate the rational functions`(2x)/((x^2+1)(x^2+3)`

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Let `x^2=t⇒2xdx=dt`
∴`∫2x/((x^2+1)(x^2+3))dx=∫dt/((t+1)(t+3)) .........(1)`
Let `1/((t+1)(t+3))​=A/(t+1)+B/(t+3)`
⇒`1=A(t+3)+b(t+1) ......... (1)`
Substituting t=−3 and t=−1 in equation (1) we obtain
`A=1/2 and B=−1/2`
∴`1/((t+1)(t+3))​=1/((t+1)*2)​−1/((t+3)*2)`
⇒`∫2x/((x^2+1)(x^2+3))dx=∫{1/((t+1)*2)−1/((t+3)*2)​}dt`
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