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प्रश्न
Evaluate the definite integral:
`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx
उत्तर
Let `I = int_0^(pi/4) (sinx cos x)/(sin^4x + cos^4) dx`
Dividing numerator and denominator by cos4 x
`I = int_0^(pi/4) (tan x sec^2x)/ (1 + tan^4) dx`
Put tan2 x = t
⇒ 2 tanx sec2x dx = dt
When x = 0, t = 0 and `x = pi/4` , t = 1
∴ `I = 1/2 int_0^1 dt/ (1 + t^2)`
`= [1/2 tan^-1 t]`
`= 1/2 xx pi/4 `
`= pi/8`
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