Indefinite Integration
Trigonometric Functions
Grade 12
Question:
<p>\(\int \frac{\tan x}{\sin x \cos x} dx\) is equal to [IIT - 1998]</p>
<p>(A) \(\sin x \log\tan\frac{x}{2} + c\)</p>
<p>(B) \(\sin x \log\tan\frac{x}{2} - x + c\)</p>
<p>(C) \(\sin x \log\frac{\tan\frac{x}{2}}{2} + x + c\)</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Simplify the trigonometric expression and use substitution with logarithmic integration techniques.
<p>Rewrite $\tan x = \frac{\sin x}{\cos x}$, so the integral becomes $\int \frac{\sin^2 x}{\sin x \cos^2 x} dx = \int \frac{\sin x}{\cos^2 x} dx$. Use substitution and partial fractions with logarithmic identities to get the result.</p>
Correct Answer: B