Integrate 1/ (1 + sin x)?

Solution:

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) / cos²x dx

= ∫ (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.

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Integrate 1/ (1 + sin x)?

Summary:

The integral of 1/ (1 + sin x) is ∫1/(1 + sin x) dx = tan x − sec x + C