Question

Using properties of the definite integral, calculate the value of: 

`∫_0^(x/2) sin^2 x/(1 + sinx cos x) dx`

Answer 1

Let 

I = `∫_0^(x/2) sin^2 x/(1 + sinx cos x) dx`    ...[1]

`∫_0^a f(x)dx = ∫_0^a f(a − x)dx`

Replacing x with (`π/2` − x)

I = `∫_0 ^(π/2) cos^2 x/(1 + cos x sin x) dx`   ...[2]

2I = `∫_0 ^(π/2) (sin^2 x + cos^2 x)/(1 + sin x cos x) dx`    ...[Adding (1) and (2)]

= `∫_0 ^(π/2) 1/(1 + sin x cos x) dx`

2I = `∫_0 ^(π/2) sec^2 x/(sec^2 x + tan x) dx = ∫_0 ^(π/2) sec^2 x/(1 + tan^2 x + tan x) dx`

Let tan x = t, then sec² x dx = dt

limits change from [0, π/2] to [0, ∞].

2I = `∫_0^∞ dt/(t^2 + t +1) = ∫_0^∞ dt/((t + 1/2)^2 + (sqrt3/2)^2)`

`∫ dx/(x^2 + a^2) = 1/a tan^(−1) (x/a)`

2I `[2/sqrt3 tan^(−1)] ((t + 1/2)/(sqrt3/2))]_0^∞`

2I = `2/sqrt3[tan^(−1)(∞) − tan^(−1)(1/sqrt3)]`

2I = `2/sqrt3[π/2 − π/6]`

2I = `2/sqrt3 × π/3`

2I = `(2π)/(3sqrt3)`

I = `π/(3sqrt3)`