Let ∫ 3 + t a n x 2 − t a n x d x = 2 1 ( αx + lo g e ∣ β sin x + γ cos x ∣ ) + C , where C is the constant of integration. Then α + β γ is equal to :

Question

Let ∫ 3 + t a n x 2 − t a n x ​ d x = 2 1 ​ ( αx + lo g e ​ ∣ β sin x + γ cos x ∣ ) + C , where C is the constant of integration. Then α + β γ ​ is equal to :

TestHub · Accepted Answer

​ ∫ 3 + tan x 2 − tan x ​ d x = ∫ 3 cos x + sin x 2 cos x − sin x ​ d x 2 cos x − sin x = A ( 3 cos x + sin x ) + B ( cos x − 3 sin x ) 3 A + B = 2 A − 3 B = − 1 ​ ​ ⇒ A = 2 1 ​ , B = 2 1 ​ ∴ ∫ 3 cos x + sin x 2 cos x − sin x ​ d x = 2 x ​ + 2 1 ​ ln ∣3 cos x + sin x ∣ + C = 2 1 ​ ( x + ln ∣3 cos x + sin x ∣ ) + C = 2 1 ​ ( αx + ln ∣ β sin x + γ cos x ∣ ) + C α = 1 , β = 1 , γ = 3 ∴ α + β γ ​ = 1 + 1 3 ​ = 4 ​

Mathematics - Indefinite Integration Question with Solution | TestHub

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