TargetIIT

(1) Let [im]http://codecogs.izyba.com/gif.latex?I%3D%5Cint_%7B1/2%7D%5E%7B2%7D%7B%5Cfrac%7Bln%28t%29%7D%7B1+t%5E%7Bn%7D%7D%7Ddt[/im],
find the sign of Integral for different values of [im]http://codecogs.izyba.com/gif.latex?n%5Cvarepsilon%20N%20+%5Cleft%5C%7B0%20%5Cright%5C%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?I%20%3D%20%5Cint_%7B%5Cfrac%7B1%7D%7B2%7D%7D%5E2%20%5Cfrac%7B%5Cln%20t%7D%7B1+t%5En%7D%20%5C%20dt%20%3D%20%5Cint_%7B%5Cfrac%7B1%7D%7B2%7D%7D%5E1%20%5Cfrac%7B%5Cln%20t%7D%7B1+t%5En%7D%20%5C%20dt%20+%20%5Cint_%7B1%7D%5E2%20%5Cfrac%7B%5Cln%20t%7D%7B1+t%5En%7D%20%5C%20dt[/im]

[im]http://codecogs.izyba.com/gif.latex?%3D%20%5Cint_%7B%5Cfrac%7B1%7D%7B2%7D%7D%5E1%20%5Cfrac%7B%5Cln%20t%7D%7B1+t%5En%7D%20%5C%20dt%20-%20%5Cint_%7B%5Cfrac%7B1%7D%7B2%7D%7D%5E1%5Cfrac%7Bt%5E%7Bn-2%7D%20%5Cln%20t%7D%7B1+t%5En%7D%20%5C%20dt%20%3D%20%5Cint_%7B%5Cfrac%7B1%7D%7B2%7D%7D%5E1%5Cfrac%7B%281-t%5E%7Bn-2%7D%29%20%5Cln%20t%7D%7B1+t%5En%7D%20%5C%20dt[/im]

We have ln t<0 for 0<t<1

Also for n>2 [im]http://codecogs.izyba.com/gif.latex?1-t%5E%7Bn-2%7D%3E0[/im] in this range and for n≤2 [im]http://codecogs.izyba.com/gif.latex?1-t%5E%7Bn-2%7D%3C0[/im]

Thus we see that the integral is negative for n>2 and +ve for n≤2

I am unable to edit the post, it should be +ve for n=0,1 and -ve otherwise