TS EAMCET · Maths · Indefinite Integration
\(\int \frac{d x}{4 \sin x+3 \cos x}=\)
- A \(\frac{1}{5} \log \left|\sec \left(x-\tan ^{-1} \frac{4}{3}\right)\right|+c\)
- B \(\frac{1}{5} \log \left|\tan \left(\frac{\pi}{4}-x+\tan ^{-1} \frac{4}{3}\right)\right|+c\)
- C \(\frac{1}{5} \log \left|\sec \left(x-\tan ^{-1} \frac{4}{3}\right)+\tan \left(x-\tan ^{-1} \frac{4}{3}\right)\right|+c\)
- D \(\frac{1}{5} \log \left|\operatorname{cosec}\left(x-\tan ^{-1} \frac{4}{3}\right)+\cot \left(x-\tan ^{-1} \frac{4}{3}\right)\right|+c\)
Answer & Solution
Correct Answer
(C) \(\frac{1}{5} \log \left|\sec \left(x-\tan ^{-1} \frac{4}{3}\right)+\tan \left(x-\tan ^{-1} \frac{4}{3}\right)\right|+c\)
Step-by-step Solution
Detailed explanation
\(I=\int \frac{d x}{4 \sin x+3 \cos x}\) Let \(3=r \cos \theta\) and \(4=r \sin \theta\) \(\Rightarrow \quad r=5\) and \(\theta=\tan ^{-1} \frac{4}{3}\) So, \(I=\int \frac{d x}{r \cos (x-\theta)}=\frac{1}{5} \int \sec (x-\theta) d \theta\)…
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