TS EAMCET · Maths · Indefinite Integration
\(\int \frac{d x}{7+5 \cos x}\) is equal to
- A \(\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{1}{\sqrt{3}} \tan \frac{x}{2}\right)+c\)
- B \(\frac{1}{\sqrt{6}} \tan ^{-1}\left(\frac{1}{\sqrt{6}} \tan \frac{x}{2}\right)+c\)
- C \(\frac{1}{7} \tan ^{-1}\left(\tan \frac{x}{2}\right)+c\)
- D \(\frac{1}{4} \tan ^{-1}\left(\tan \frac{x}{2}\right)+c\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{\sqrt{6}} \tan ^{-1}\left(\frac{1}{\sqrt{6}} \tan \frac{x}{2}\right)+c\)
Step-by-step Solution
Detailed explanation
\text { Let } \begin{aligned} I & =\int \frac{d x}{7+5 \cos x} \\ & =\int \frac{d x}{7\left(\cos ^2 \frac{x}{2}+\sin ^2 \frac{x}{2}\right)+5\left(\cos ^2 \frac{x}{2}-\sin ^2 \frac{x}{2}\right)} \\ & =\int \frac{d x}{12 \cos ^2 \frac{x}{2}+2 \sin ^2 \frac{x}{2}} \\ & =\frac{1}{2}…
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