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Question

Integrate the rational functions.
2x(x2+1)(x2+3)dx.

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Solution

Let I=2x(x2+1)(x2+3)dx
Put x2=t2x=dtdxdx=dt2x
I=2x(t+1)(t+3)dt2x=1(t+1)(t+3)dt
Let 1(t+1)(t+3)=At+1+Bt+3=At+3A+Bt+B(t+1)(t+3)
1=(A+B)t+(3A+B)
On comparing the coefficients of t and constant terms on both sides, we get
A+B=0 .....(i)
and 3A+B =1 ....(ii)
On subtracting Eq. (ii)from Eq. (i), we get
2A=1A=12
On putting the value of A in Eq. (i), we get B=12
I=121t+1dt121t+3dt=12log|t+1|12log|t+3|+C=12logt+1t+3+C=12logx2+1x2+3+C [Put t=x2]


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