Question

If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.

Options

• 3

• 4

• `1/3`

• `1/4`

Answer 1

If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is `underline(1/4)`.

Explanation:

`int_0^"k" "dx"/(2 + 32x^2) = 1/32 int_0^"k" "dx"/(x^2 + 1/16)`

`=> pi/24 = 1/32 int_0^"k" "dx"/(x^2 + (1/4)^2)`

`= 1/32 * 1/(1/4) [tan^-1 x/(1/4)]_0^"k"`

`= 1/8 [tan^-1 "4x"]_0^"k"`

`=> pi/32 = 1/8 (tan^-1 "4k" - 0)`

`=> pi/4 = tan^-1 "4k"`

`=> tan pi/4` = 4k

`=> 4"k" = 1`

`=> "k" = 1/4`