EXPLANATION.

⇒ ∫sin(x)/sin(x - a)dx.

As we know that,

By using substitution method, we get.

⇒ x - a = t.

Differentiate w.r.t x, we get.

⇒ dx = dt.

⇒ x = t + a.

Put the value in the equation, we get.

⇒ ∫sin(t + a)/sin(t)dt.

As we know that,

FORMULA of :

⇒ sin(A ± B) = sin(A).cos(B) ± cos(A).sin(B).

Using this formula in equation, we get.

⇒ ∫sin(t).cos(a) + cos(t).sin(a)/sin(t) dt.

⇒ ∫sin(t).cos(a)/sin(t) dt + ∫cos(t).sin(a)/sin(t) dt.

⇒ ∫cos(a)dt + ∫cot(t).sin(a)dt.

⇒ cos(a) ∫dt + sin(a) ∫cot(t)dt.

⇒ cos(a)(t) + sin(a) sin(t) + c.

Put the value of t = (x - a) in equation, we get.

⇒ cos(a)[x - a] + sin(a) sin[x - a] + c.

MORE INFORMATION.

Standard INTEGRALS.

(1) = ∫sin x dx = - cos(x) + c.

(2) = ∫cos x dx = sin(x) + c.

(3) = ∫tan x dx = ㏒(sec x) + c = -㏒(cos x) + c.

(4) = ∫cot x dx = sin x + c.

(5) = ∫sec x dx = ㏒(sec x + tan x) + c = -㏒(sec x - tan x) + c = ㏒ tan(π/4 + x/2) + c.

(6) = ∫cosec x dx = -㏒(cosec x + cot x) + c = ㏒(cosec x - cot x) + c = ㏒ tan(x/2) + c.

(7) = ∫sec x tan x dx = sec x + c.

(8) = ∫cosec x cot x dx = -cosec x + c.

(9) = ∫sec²xdx = tan x + c.

(10) = ∫cosec²xdx = -cot x + c.