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प्रश्न
Evaluate the following : `int_((-pi)/4)^(pi/4) (x + pi/4)/(2 - cos 2x)*dx`
उत्तर
Let I = `int_((-pi)/4)^(pi/4) (x + pi/4)/(2 - cos 2x)*dx`
= `int_((-pi)/4)^(pi/4) [x/(2 - cos2x) + (pi/4)/(2 - cos 2x)]`
= `int_((-pi)/4)^(pi/4) x/(2 - cos2x)*dx + pi/(4) int_((-pi)/4)^(pi/4) 1/(2 - cos2x)*dx`
= `"I"_1 + pi/(4)"I"_2` ...(1)
Let f(x) = `x/(2 - cos2x)`
∴ f(– x) = `(-x)/(2 - cos[2(-x)]`
= `(-x)/(2 - cos 2x)`
= – f(x)
∴ f is an odd function
∴ `int_((-pi)/4)^(pi/4) f(x)*dx` = 0
i.e. `int_((-pi)/4)^(pi/4) x/(2 - cos 2x)*dx` = 0, i.e. I1 = 0 ...(2)
In I2, put tan x = t
∴ x = tan–1t
∴ dx = `(1)/(1 + t^2)*dt`
and
cos 2x = `(1 - t^2)/(1 + t^2)`
When x = `- pi/(4), t = tan(- pi/4)` = – 1
When x = `pi/(4), t = tan pi/(4)` = 1.
∴ I2 = `int_(-1)^(1) (1)/(2 - ((1 - t^2)/(1 + t^2)))*(1)/(1 + t^2)*dt`
= `int_(-1)^(1) (1)/(2(1 + t^2) - (1 - t^2))*dt`
= `int_(-1)^(1) (1)/(3t^2 + 1)*dt`
= `int_(-1)^(1) (1)/((sqrt(3) t)^2 + 1)`
= `[1/sqrt(3) tan^-1 ((sqrt(3)t)/1)]_(-1)^(1)`
= `(1)/sqrt(3)[tan^-1 sqrt(3) - tan^-1 (- sqrt(3))]`
= `(1)/sqrt(3)[tan^-1 sqrt(3) + tan^-1 sqrt(3)]`
= `(1)/sqrt(3)[pi/3 + pi/3]`
= `(2pi)/(3sqrt(3)` ...(3)
From (1), (2) and (3), we get
I = `0 + pi/(4)[(2pi)/(3sqrt(3))]`
= `pi^2/(6sqrt(3)`.
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