If ∫ π 6 π 3 1 - sin 2 x d x = α + β 2 + γ 3 , where α , β and γ are rational numbers, then 3 α + 4 β - γ is equal to _____.

Step-by-step solution

Given, ∫ π 6 π 3 1 - sin 2 x d x = α + β 2 + γ 3 Now, let I = ∫ π 6 π 3 1 - sin 2 x d x ⇒ I = ∫ π 6 π 3 | sin x - cos x | d x ⇒ I = ∫ π 6 π 4 ( cos x - sin x ) d x + ∫ π 4 π 3 ( sin x - cos x ) d x ⇒ I = - 1 + 2 2 - 3 Now, comparing with I = α + β 2 + γ 3 We get, α = - 1 , β = 2 , γ = - 1 Hence, 3 α + 4 β - γ = - 3 + 8 + 1 = 6