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Putcot-1(ex)=zOr,ex=cotzor,exdx=-cosec2zdzNow multiply the numerator and denominator by ex.It will becomeNow replace all x,dx by the values of z,dz.It will becomeNow we shall carry on integration by parts taking z as 1st function.Now we have to replace z,tanz,cosz in terms of x.z=cot-1(ex)tanz=1/cotz=1/exAlso we know that1+cos2z=2cos2zor,cos2z=(cos2z+1)/2Put these all together and the answer will become
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