∫(1 +sin x /1- sin x )dx
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EXPLANATION.
⇒ ∫(1 + sin x/1 - sin x)dx.
As we know that,
Rationalizes the equation, we get.
⇒ ∫(1 + sin x)/(1 - sin x) X (1 + sin x)/(1 + sin x) dx.
⇒ ∫(1 + sin x)²/(1 - sin x)(1 + sin x) dx.
⇒ ∫(1 + sin x)²/(1 - sin²x) dx.
⇒ ∫(1 + sin x)²/cos²x dx.
⇒ ∫(1 + sin²x + 2sin x)/cos²x dx.
⇒ ∫1/cos²x dx + ∫(sin²x/cos²x) dx + ∫(2sinx/cos²x) dx.
⇒ ∫sec²x dx + ∫tan²xdx + ∫2secx tan x dx.
⇒ ∫sec²x dx + ∫(sec²x - 1)dx + 2∫sec x tan x dx.
⇒ ∫sec²x dx + ∫sec²x dx - ∫dx + 2∫sec x tan x dx.
⇒ tan x + tan x - x + 2 sec x + c.
⇒ 2tanx - x + 2sec x + c.
MORE INFORMATION.
(1) ∫sin x dx = - cos x + c.
(2) ∫sin (ax + b)dx = - cos (ax + b)/a + c.
(3) ∫cos x dx = sin x + c.
(4) ∫sec²x dx = tan x + c.
(5) ∫cosec²x dx = - cot x + c.
(6) ∫sec x tan x dx = sec x + c.
(7) ∫cosec x cot x dx = - cosec x + c.
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