Question

`int_(1/a)^a(tan^-1x)/xdx=?`

Answer 1

Let
`z=lnx=>x=e^z`
`x=a->z=lna and x=1/a->z=-lna`

`and dz==(dx)/x`

`I=int_(1/a)^a(tan^-1x)/xdx`

`color(red)("Replacing "a" by "lna , 1/a" by "-lna,x" by "e^z and {dx}/x" by "dz)`

`I=int_(-lna)^(lna)tan^-1(e^z)dz`

`=int_(-lna)^0tan^-1e^zdz+int_0^(lna)tan^-1e^zdz`

`=int_0^(lna)tan^-1e^-zdz+int_0^(lna)tan^-1e^zdz`

`=int_0^(lna)(cot^-1e^z+tan^-1e^z)dz`

`color(blue)("Applying formula " tan^-1x+cot^-1x=pi/2)`

`=int_0^(lna)(pi/2)dz`

`=pi/2lna`