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