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Question

x2+1x2-1 dx

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Solution

x2+1x2-1dx=x2-1+2x2-1dx=dx+21x2-12dx=dx+21x-1 x+1dx ... 1 1x-1x+1=Ax-1+Bx+11x-1 x+1=A x+1+Bx-1x-1 x+11=A x+1 B x-1 ....(2)Putting x+1=0 or x=-1 in eq (2)1=A×0+B -1 -1B=-12Putting x-1=0 or x=1 in eq (2)1=A 1+1+B×0A=12 1x-1x+1=12x-1-12x+1 ...(3)From eq (1) & eq (3) x2+1x2-1dx=dx+212 x-1-12 x+1dx =dx+dxx-1-dxx+1 =x+ln x-1=-ln x+1+C =x+ln x-1x+1+C

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