Question

If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.

पर्याय

• 3

• 4

• `1/3`

• `1/4`

Answer 1

If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is `underlinebb(1/3)`.

Explanation:

Here, `int_0^k dx/(2 + 18x^2) = π/24`

`\implies π/24 = 1/18 int_0^k dx/((1/9) + x^2) = 1/18int_0^k dx/((1/3)^2 + x^2)`

`\implies π/24 = 1/18 xx 1/((1/3)) tan^-1[x/((1/3))]_0^k`

`\implies π/24 = 3/18 tan^-1[3x]_0^k = 1/6[tan^-1  3k - tan^-1 0]`

`\implies π/24 = 1/6[tan^-1 3k - 0]`

∴ `(6π)/24` = tan^–1 3k

`\implies tan  π/4` = 3k

`\implies` 1 = 3 K

`\implies` K = `1/3`