TS EAMCET · Maths · Indefinite Integration
If \(\int \frac{a \cos x-2 \sin x}{b \sin x+5 \cos x} d x=\frac{7}{41} x+\frac{22}{7} \log\) \(|b \sin x+5 \cos x|+C,(a>0, b>0)\), then \(\int \frac{d x}{b+a \cos x}=\)
- A \(\frac{2}{3} \log \left(\frac{3 \tan \frac{x}{2}+4-\sqrt{3}}{3 \tan \frac{x}{2}+4+\sqrt{3}}\right)+C\)
- B \(\frac{2}{\sqrt{7}} \tan ^{-1}\left(\frac{\tan \frac{x}{2}}{\sqrt{7}}\right)+C\)
- C \(\frac{2}{\sqrt{7}} \log \left(\frac{\sqrt{7}-\tan \frac{x}{2}}{\sqrt{7}+\tan \frac{x}{2}}\right)+C\)
- D \(2 \sinh ^{-1}\left(\frac{2 \tan \frac{x}{2}}{1+\tan ^2 \frac{x}{2}}\right)+C\)
Answer & Solution
Correct Answer
(B) \(\frac{2}{\sqrt{7}} \tan ^{-1}\left(\frac{\tan \frac{x}{2}}{\sqrt{7}}\right)+C\)
Step-by-step Solution
Detailed explanation
We have, \(\int \frac{a \cos x-2 \sin x}{b \sin x+5 \cos x} d x=\frac{7}{41} x+\frac{22}{41}\) \(\begin{aligned} & \log |(b \sin x+5 \cos x)|+C \\ & \text { Now, let } a \cos x-2 \sin x=\lambda(b \sin x+5 \cos x) \\ & \qquad+\mu(b \cos x-5 \sin x)\end{aligned}\)…
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