Question

Evaluate I = `int_0^1 int_0^(sqrt(1+x^2)) (dx.dy)/(1+x^2+y^2)`

Answer 1

I = `int_0^1 int_0^(sqrt(1+x^2)) (dx.dy)/(1+x^2+y^2)`

I= `int_0^1 1/(sqrt(1+x^2)) [tan^(-1)  y/sqrt(1+x^2)]_0^sqrt(1+x^2) dx` 

`therefore "I" = int_0^1 pi/4 1/sqrt(1+x^2) dx `

`therefore "I"=pi/4[log(x+sqrt(1+x^2))]_0^1`

`therefore  "I" =pi/4log(1+sqrt2)`