"Evaluate the integral. ∫(1−sin⁡x)tan⁡xdx" was answered by students like you

kolonelyf4

Answered question

2021-12-03

Evaluate the integral.

\int \frac{(1 - \sin x)}{\tan x} d x

Robert Harris

Beginner2021-12-04Added 23 answers

Step 1
Consider the given integration as

\int \frac{(1 - \sin x)}{\tan x} d x

I = \int \frac{1}{\tan x} d x - \int \frac{\sin x}{\tan x} d x

Step 2
Now

\int \frac{1}{\tan x} d x = \int \cot x d x

= - \ln (\csc x) + c

= \ln (\frac{1}{\csc x}) + c

= \ln (\sin x) + c

And

\int \frac{\sin x}{\tan x} d x = \int \frac{\sin x}{\frac{\sin x}{\cos x}} d x

= \int \cos x d x

= \sin x + c

Step 3
Thus

I = \ln (\sin x) + c - \sin x + c

I = \ln (\sin x) - \sin x + C

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