Evaluate : ∫(sin x dx/(sin x + cos x)), x ∈ [0,π/2]

1.	Evaluate : ∫(sin x dx/(sin x + cos x)), x ∈ [0,π/2]
Answer» ∫(sin x dx/(sin x + cos x)), x ∈ [0,π/2] Let I = ∫(sin x/(sin x + cos x)) dx, x ∈ [0,π/2] ...(i) I = ∫(sin((π/2) - x)/(sin((π/2) - x) + cos((π/2) - x)) dx, x ∈ [0,π/2] I = ∫(cos x/cos x + sin x)) dx, x ∈ [0,π/2] ...(ii) Adding eq. (i) and (ii), 2I = ∫((sin x + cos x)/(sin x + cos x)) dx, x ∈ [0,π/2] 2I = ∫1.dx, x ∈ [0,π/2] 2I = [x], x ∈ [0,π/2] 2I = (π/2) - 0 I = π/4

Answer»

∫(sin x dx/(sin x + cos x)), x ∈ [0,π/2]

Let I = ∫(sin x/(sin x + cos x)) dx, x ∈ [0,π/2] ...(i)

I = ∫(sin((π/2) - x)/(sin((π/2) - x) + cos((π/2) - x)) dx, x ∈ [0,π/2]

I = ∫(cos x/cos x + sin x)) dx, x ∈ [0,π/2] ...(ii)

Adding eq. (i) and (ii),

2I = ∫((sin x + cos x)/(sin x + cos x)) dx, x ∈ [0,π/2]

2I = ∫1.dx, x ∈ [0,π/2]

2I = [x], x ∈ [0,π/2]

2I = (π/2) - 0

I = π/4

Discussion