AP EAMCET · Maths · Indefinite Integration
\(\int \frac{d x}{4+5 \cos x}=\)
- A \(-\frac{1}{3} \log \left|\frac{3+\tan \frac{x}{2}}{3-\tan \frac{x}{2}}\right|+C\)
- B \(\frac{1}{3} \log \left|\frac{3+\tan \frac{x}{2}}{3-\tan \frac{x}{2}}\right|+C\)
- C \(-\frac{1}{9} \log \left|\frac{3-\tan \frac{x}{2}}{3+\tan \frac{x}{2}}\right|+C\)
- D \(\frac{1}{9} \log \left|\frac{3-\tan \frac{x}{2}}{3+\tan \frac{x}{2}}\right|+C\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{3} \log \left|\frac{3+\tan \frac{x}{2}}{3-\tan \frac{x}{2}}\right|+C\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \text {Let } I=\int \frac{1}{4+5 \cos x} d x \\ & =\int \frac{1}{4+5\left(\frac{1-\tan ^2 x / 2}{1+\tan ^2 x / 2}\right)} d x \end{aligned}\) Let \(u=\tan \frac{x}{2} \Rightarrow \frac{d u}{d x}=\frac{\sec ^2 \frac{x}{2}}{2} d u\)…
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