Integrate 1/ (1 + sin x)?

We need to find the integral of 1/ (1 +sin x)


1/ (1 +sin x)

Multiplying numerator and deniminator by (1 – sin x) we get

\(\int \frac{1}{1 + sin x}dx\)

= \(\int \frac{1-sin x}{1-sin^{2}x} dx\)

from trigonometric identity we know that sin2x + cos2x = 1

cos2x = 1 – sin2x

= \(\int \frac{1-sin x}{cos^{2}x} dx\)

=\(\int \sec ^{2}x – \tan x \sec xdx\)

=\(\tan x- \sec x . dx +c\)


\(\int \frac{1}{1 + sin x}dx\) =\(\tan x- \sec x . dx +c\)

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