Question

Evaluate the definite integral:

`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx

Answer 1

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`