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प्रश्न
Evaluate : `int_0^(pi/4) (cosx)/(4 - sin^2x)*dx`
उत्तर
Let I = `int_0^(pi/4) (cosx)/(4 - sin^2x)*dx`
Put sin x = t
∴ cos x·dx = dt
When x = `pi/(4), t = sin pi/(4) = (1)/sqrt(2)`
When x = 0, t = sin 0 = 0.
∴ I = `int_0^(1/sqrt(2)) dt/(2^2 - t^2)`
= `[1/(2(2)) log|(2 + t)/(2 - t)|]_0^((1)/sqrt(2))`
= `(1)/(4)[log((2 + 1/sqrt(2))/(2 - 1/sqrt(2))) - log((2 + 0)/(2 - 0))]`
= `(1)/(4)[log((2sqrt(2) + 1)/(2sqrt(2) - 1)) - log 1]`
= `(1)/(4)log((2sqrt(2) + 1)/(2sqrt(2) - 1))`. ...[∵ log 1 = 0]
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