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Question

Integrate the rational functions.
x3+x+1x21dx.

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Solution

x3+x+1x21dx=xdx+2x+1x21dx[x3+x+1x21=x+2x+1x21]=xdx+2x+1(x+1)(x1)dx
Let 2x+1(x+1)(x1)=A(x+1)+B(x1)
2x+1(x+1)(x1)=A(x1)+B(x+1)(x+1)(x1)2x+1=x(A+B)A+B
On comparing the coefficients of x and constant of x and constant term on both sides we get
A+B=2 and A+B=1
On Adding above equations, we get 2B=3B=32 and then A=12
x2+x+1x21dx=xdx+A(x+1)dx+B(x1)dx=xdx+121(x+1)dx+321(x1)dx=x22+12log|x+1|+32log|x1|+C


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