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प्रश्न
Evaluate the following : `int_0^(pi/4) (tan^3x)/(1 +cos2x)*dx`
उत्तर
Let I = `int_0^(pi/4) (tan^3x)/(1 +cos2x)*dx`
= `int_0^(pi/4) (tan^3x)/(2cos^2x)*dx`
= `(1)/(2) int_0^(pi/4) tan^3x*sec^2x*dx`
Put tan x = t
∴ sec2x·dx = dt
When x = 0, t = tan 0 = 0
When x = `pi/(4), t = tan pi/(4)` = 1
∴ I = `(1)/(2) int_0^1 t^3*dt`
= `(1)/(2)*[(t^4)/4]_0^1`
= `(1)/(8)[t^4]_0^1`
= `(1)/(8)[1 - 0]`
= `(1)/(8)`.
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