Question

Find the value of \(\int \frac{1 - \sin x}{\cos^2 x} \, dx\)

• A. \(\tan x + \sec x + \text{constant}\)
• B. \(\sec x - \tan x + \text{constant}\)
• C. \(\tan x - \sec x + \text{constant}\)
• D. \(-\tan x - \sec x + \text{constant}\)

Hint:

Remember: \(\frac{d}{dx}(\tan x - \sec x) = \sec^2 x - \sec x \tan x = \frac{1 - \sin x}{\cos^2 x}\). Differentiating the options is a quick check in MCQs.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Trigonometric integrals involving algebraic combinations in the numerator over a single denominator term can be split into separate elementary trigonometric integrals.
Key Formula or Approach:
\[ \int \sec^2 x \, dx = \tan x + C \]
\[ \int \sec x \tan x \, dx = \sec x + C \]

Step 2: Detailed Explanation:
Splitting the integrand into two fractions:
\[ \int \frac{1 - \sin x}{\cos^2 x} \, dx = \int \left( \frac{1}{\cos^2 x} - \frac{\sin x}{\cos^2 x} \right) dx \]
Expressing in terms of standard trigonometric functions:
\[ = \int \left( \sec^2 x - \sec x \tan x \right) dx \]
Integrating each term individually:
\[ = \int \sec^2 x \, dx - \int \sec x \tan x \, dx \]
\[ = \tan x - \sec x + \text{constant} \]

Step 3: Final Answer:
Thus, the evaluated integral is \(\tan x - \sec x + \text{constant}\), matching option (C).