Find the most general antiderivative.
-$$\int \frac { \sec \theta } { \sec \theta - \cos \theta } d \theta$$
A) $\cos ^ { 2 } \theta + C$
B) $\cot \theta + \mathrm { C }$
C) $\theta + \tan \theta + \mathrm { C }$
D) $- \cot \theta + C$

Question

Quizplus · Accepted Answer

First, we use the identity $\sec 	heta - \cos 	heta = \frac{\sin 	heta}{\cos 	heta}$. Then we can rewrite the integral as:
$$\int \frac{\sec 	heta}{\sec 	heta - \cos 	heta} d	heta = \int \frac{\sec 	heta \cos 	heta}{\sin 	heta} d	heta = \int \frac{d}{d	heta} \ln|\sin 	heta| d 	heta = \ln|\sin 	heta| + C$$
Using logarithmic differentiation, we get:
$$\frac{d}{d	heta} \ln|\sin 	heta| = \frac{1}{|\sin 	heta|} \frac{d}{d	heta} |\sin 	heta| = \frac{\cos 	heta}{\sin 	heta}$$
Therefore, the most general antiderivative is $\ln|\sin	heta|+C$, which can be rewritten as $-\ln|\cos	heta|+C=-\log|\cos	heta|+C=-\cot	heta+C$ (using the properties of logarithms). Thus the answer is D.

Find the Most General Antiderivative ∫sec⁡θsec⁡θ−cos⁡θdθ\int \frac { \sec \theta } { \sec \theta - \cos \theta } d \theta∫secθ−cosθsecθ​dθ

Find the Most General Antiderivative $\int \frac { \sec \theta } { \sec \theta - \cos \theta } d \theta$