If f(x)=limn→∞ xn−x−nxn+x−n,x>1  then ∫xf(x)log⁡x+1+x21+x2dx is

f′(x)=limn→∞ xn−x−nxn+x−n=limn→∞ 1−(1/x)2n1+(1/x)2n=1(1/x<1) Using integration by parts , we get I=∫xlog⁡x+1+x21+x2dx=1+x2log⁡x+1+x2dx−∫dx=1+x2log⁡x+1+x2−x+C.