Question:
Evaluate: \( \int \frac{1 + \tan x \tan(x + \alpha)}{\tan x \tan(x + \alpha)} dx \):
Evaluate: \( \int \frac{1 + \tan x \tan(x + \alpha)}{\tan x \tan(x + \alpha)} dx \):
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Use trigonometric identities and log properties to evaluate integrals involving tan functions.
Updated On: May 19, 2025
- \( \tan \alpha (\log(\sec(x + \alpha)) + \log \sec x + C) \)
- \( \cot \alpha (\log |\sin x| - \log |\sin(x + \alpha)|) + C \)
- \( \tan \alpha \left( \log \frac{\cos x}{\sin(x + \alpha)} \right) + C \)
- \( \cot \alpha \left( \log \frac{\sin(x + \alpha)}{\cos(x + \alpha)} \right) + C \)
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The Correct Option is B
Solution and Explanation
Break the integrand into partial fractions and simplify. Use identities to convert to log form. Resultant expression simplifies to:
\[
\cot \alpha (\log |\sin x| - \log |\sin(x + \alpha)|) + C
\]
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