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Question
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
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Solution
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = `1/2`.
Explanation:
Given that `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`
⇒ `1/4 int_0^"a" 1/((1/4 + x^2)) "d"x = pi/8`
⇒ `int_0^pi 1/([(1/2)^2 + x^2]) "d"x = pi/2`
⇒ `1/(1/2) [tan^-1 x/(1/2)]_0^"a" = pi/2`
⇒ `2[tan^-1 2"a" - tan^-1 0] = pi/2`
⇒ `tan^-1 2"a" = pi/4`
⇒ 2a = `tan pi/4`
⇒ 2a = 1
⇒ a = `1/2`.
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