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Question

a+xxdx

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Solution

Let I=a+xxdx =a+x a+xx a+x =a+xx2+axdx =a1x2+axdx+x x2+axdx =a1x2+ax+a22-a22dx+x x2+axdx =a1x+a22-a22dx+122x x2+axdx =a1x+a22-a22dx+122x+a-ax2+axdx =a1x+a22-a22dx+122x+a x2+axdx-a21x2+axdx =a21x+a22-a22dx+122x+a x2+ax dxPutting x2+ax=t in the Ist integral 2x+a dx=dt I=a21x+a22-a22dx+121tdt =a2 ln x+a2+x2+ax+12×2t+C 1x2-a2dx=lnx+x2-a2+C =a2 ln x+a2+x2+ax+x2+ax+C t=x2+ax

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