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Question

ax2+bx+cx-a x-b x-c dx, where a, b, c are distinct

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Solution

We have,I=ax2+bx+cx-a x-b x-c dxLet ax2+bx+cx-a x-b x-c=Ax-a+Bx-b+Cx-cax2+bx+c=Ax-b x-c+B x-cx-a+Cx-a x-bax2+bx+c=Ax2-b+cx+bc+Bx2-c+ax+ca+Cx2-a+bx+abax2+bx+c=A+B+Cx2-Ab+c+Bc+a+Ca+bx+Abc+Bca+CabEquating the coefficients on both sides, we geta=A+B+C .....1b=-Ab+c+Bc+a+Ca+b .....2c=Abc+Bca+Cab .....3Solving 1, 2 and 3, we getA=a3+ab+ca-ba-cB=ab2+b2+cb-ab-cC=ac2+bc+cc-ac-bI=a3+ab+ca-ba-c×1x-a+ab2+b2+cb-ab-c×1x-b+ac2+bc+cc-ac-b×1x-c dx=a3+ab+ca-ba-clog x-a+ab2+b2+cb-ab-clog x-b+ac2+bc+cc-ac-blog x-c+K

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