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 &#945; &#8712; ( 0 , &#960; 2 ) &#160; , be constant.If the integral &#8747; t a n x + t a n &#945; t a n x - tan &#945; d x = A x c o s 2 &#945; + B x s i n 2 &#945; + C , where C is a constant of integration, then the functions A ( x ) and B ( x ) are respectively

Aicharya · Accepted Answer

I = &#8747; tan x + tan &#945; tan x - tan &#945; d x = &#8747; sin x cos x + sin &#945; cos &#945; sin x c o s x - sin &#945; cos &#945; d x = &#8747; s i n x c o s &#945; + c o s x s i n &#945; s i n x c o s &#945; - c o s x s i n &#945; d x = &#8747; sin x + &#945; sin x - &#945; d x = &#8747; sin x - &#945; + 2 &#945; s i n ( x - &#945; ) d x = &#8747; sin x - &#945; c o s 2 &#945; + cos x - &#945; s i n 2 &#945; s i n ( x - &#945; ) d x = &#8747; c o s 2 &#945; + s i n 2 &#945; cot x - &#945; d x = x c o s 2 &#945; + s i n 2 &#945; log e sin x - &#945; + C = ( x - &#945; ) c o s 2 &#945; + s i n 2 &#945; log e sin x - &#945; + C Hence, A x = x - &#945; and B x = l o g e s i n x - &#945;

Let α ∈ ( 0 , π 2 )   , be constant.If the integral ∫ t a n x + t a n α t a…

Solution

Let &#945; &#8712; ( 0 , &#960; 2 ) &#160; , be constant.If the integral &#8747; t a n x + t a n &#945; t a…

Solution

Let α ∈ ( 0 , π 2 ) , be constant.If the integral ∫ t a n x + t a n α t a…