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Question

5xx+1 x2-4 dx

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Solution

5xx+1 x2-4dxLet5xx+1 x-2 x+2=Ax+1+Bx-2+Cx+25xx+1 x-2 x+2=A x-2 x+2+B x+1 x+2+C x+1 x-2x+1 x-2 x+25x=A x-2 x+2+B x+1 x+2+C x+1 x-2 ....(1)Putting x-2=0 or x=2 in eq (1)5×2=B 2+1 2+2B=103×4 =56Putting x+2=0 or x=-2 in eq (1)5×-2=C -2+1 -2-2-10-1×-4 =CC=-52Putting x+1=0 or x=-1 in eq (1) -5=A -1-2 -1+2-5-3=AA=53 5xx+1 x-2 x+2=53×1x+1+56 x-2- 52 x+25xx+1 x-2 x+2=56×2x+1+56 x-2-56 3x+2 5x x+1 x-2 x+2dx=562x+1 dx+561x-2dx-563x+2 dx =562 ln x+1+ln x-2-3 ln x+2+C =56 ln x+12+ln x-2-ln x+23+C =56 ln x+12 x-2x+23+C

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