Question

Integrate the following​

Answer 1

Step-by-step explanation:

this is simplified to 2cosx/cos²x

=2/coax

=2secx

use the formula of secx dx

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Answer 2

EXPLANATION.

⇒ ∫2cos(x)dx/(1 - sin(x))(1 + sin(x)).

As we know that,

Formula of :

⇒ (x - y)(x + y) = x² - y².

Using this formula in equation, we get.

⇒ ∫2cos(x)dx/(1 - sin²x).

As we know that,

Formula of :

⇒ sin²x + cos²x = 1.

⇒ cos²x = 1 - sin²x.

Using this formula in equation, we get.

⇒ ∫2cos(x)dx/cos²x.

⇒ ∫2 dx/cos(x).

⇒ ∫2 sec x dx.

⇒ 2∫sec x dx.

⇒ 2 log(sec x + tan x) + c.

                                                                                                                       

MORE INFORMATION.

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

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

(6) = ∫sec x tan x dx = sec x + c.

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

(8) = ∫sec²x dx = tan x + c.

(9) = cosec²x dx = - cot x + c.