Question

`int_0^oo (dx)/((x^2 + a^2)(x^2 + b^2))` is

Options

• `(piab)/(a + b)`

• `pi/(2(a + b))`

• `pi/(2ab(a + b))`

• `(pi(a + b))/(2ab)`

Answer 1

`pi/(2ab(a + b))`

Explanation:

`int_0^oo (dx)/((x^2 + a^2)(x^2 + b^2)) = 1/(b^2 - a^2) int_0^oo ((x^2 + b^2) - (x^2 + a^2))/((x^2 + a^2)(x^2 + b^2))`

= `1/(b^2 - a^2) int_0^oo [1/(x^2 + a^2) - 1/(x^2 + b^2)] dx`

= `1/(b^2 - a^2) [1/a tan^-1  x/a - 1/b tan^-1  x/b]_0^oo`

= `1/(b^2 - a^2) [pi/(2a) - pi/(2b)]`

= `pi/(2ab(a + b))`