Methods Of Integration

Hence, and

We use one of the trigonometric identities to solve this. We will multiply the numerator and denominator by of 1/ (1 + sin x) by (1 – sin x). Then we get
∫1/(1 + sinx)dx
∫1/(1 + sin x)·(1 - sin x)/(1 - sin x)dx
= ∫(1 − sin x)/(1 − sin2x)dx
From trigonometric identities, we know that sin2x + cos2x = 1.
From this, cos2x = 1 – sin2x
Substituting this in the above integral,
= ∫(1 − sin x)/cos2xdx
= ∫(1/cos2x) - (sin x)/(cos x)·(1/cosx)dx
= ∫ (sec2x – tan x sec x)dx
= tan x − sec x + C (∵ ∫ sec2x dx = tan x and ∫ tan x sec x dx = sec x)
Thus, ∫1 / (1 + sin x) dx = tan x − sec x + C, where C is the integration constant.

To evaluate the integral , we'll use the technique of substitution. Let . Then the derivative is , which implies .

Substituting into the integral, we get:

This simplifies to:

Integrating , we get:

Substituting back , the expression becomes:

Therefore, the correct answer is .

We will use the identity to simplify the integral: Now, split the integral: The first integral is: The second integral is: Thus, the integral becomes: The correct answer is: