Let α ∈ ( 0 , π 2 ) , be constant.If the integral ∫ t a n x + t a n α t a n x - tan α d x = A x c o s 2 α + B x s i n 2 α + C , where C is a constant of integration, then the functions A ( x ) and B ( x ) are respectively

Question

Let α ∈ ( 0 , π 2 ) , be constant.If the integral ∫ t a n x + t a n α t a n x - tan α d x = A x c o s 2 α + B x s i n 2 α + C , where C is a constant of integration, then the functions A ( x ) and B ( x ) are respectively, x - α and l o g e s i n x - α, x + α and l o g e c o s x - α, x + α and l o g e s i n x + α, x - α and l o g e c o s x - α

MARKS App · Accepted Answer

I = ∫ tan x + tan α tan x - tan α d x = ∫ sin x cos x + sin α cos α sin x c o s x - sin α cos α d x = ∫ s i n x c o s α + c o s x s i n α s i n x c o s α - c o s x s i n α d x = ∫ sin x + α sin x - α d x = ∫ sin x - α + 2 α s i n ( x - α ) d x = ∫ sin x - α c o s 2 α + cos x - α s i n 2 α s i n ( x - α ) d x = ∫ c o s 2 α + s i n 2 α cot x - α d x = x c o s 2 α + s i n 2 α log e sin x - α + C = ( x - α ) c o s 2 α + s i n 2 α log e sin x - α + C Hence, A x = x - α and B x = l o g e s i n x - α

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