Question

What is `int_1^(e^(pi/4)) 4/(x(1+(lnx)^2))dx`?

Answer 1

`4tan^-1(pi/4)`

Explanation:

`I=int_1^(e^(pi/4))4/(x(1+(lnx)^2))dx`
Let, `lnx=u=>1/x*dx=du`
`x=1=>u=ln1=>u=0and`
`x=e^(pi/4)=>u=lne^(pi/4)=>u=pi/4`
`:.I=int_0^(pi/4)4/(1+u^2)du=[4tan^-1u]_0^(pi/4)`
`:.I=4[tan^-1(pi/4)-tan^-1(0)]=4[tan^-1(pi/4)-0]`
`:.I=4tan^-1(pi/4)`