Question

Evaluate the following : If `int_0^k 1/(2 + 8x^2)*dx = pi/(16)`, find k

Answer 1

Let I = `int_0^k 1/(2 + 8x^2)*dx`

= `(1)/(8) int_0^k 1/(x^2 + (1/2)^2)*dx`

= `(1)/(8) xx (1)/((1/2))[tan^-1 (x/((1/2)))]_0^k`

= `(1)/(4)[tan^-1 2x]_0^k`

= `(1)/(4)[tan^-1 2k - tan^-1 0]`

= `(1)/(4) tan^-1 2k`

∴ I = `pi/(16)  "gives" (1)/(4) tan^-1 2k = pi/(16)`

∴ `tan^-1 2k = pi/(4)`

∴ 2k = `tan  pi/(4)` = 1

∴ k = `(1)/(2)`.