TS EAMCET · Maths · Indefinite Integration
\(\int \frac{3 \sin x+5 \cos x+4}{\sin x+\cos x+2} d x=\)
- A \(\log (\sin x+\cos x+2)+4 x-4 \tan ^{-1}\left(1+\tan \frac{x}{2}\right)+c\)
- B \(\log (\sin x+\cos x+2)+4 x-4 \sqrt{2} \tan ^{-1}\left(\frac{1+\tan \frac{x}{2}}{\sqrt{2}}\right)+c\)
- C \(4 \log (\sin x+\cos x+2)+x-4 \sqrt{2} \tan ^{-1}\left(\frac{1+\tan \frac{x}{2}}{\sqrt{2}}\right)+c\)
- D \(\begin{aligned} & 4 \log (\sin x+\cos x+2)+4 x-4 \sqrt{2} \tan ^{-1} \ & \left(\frac{1-\tan \frac{x}{2}}{\sqrt{2}}\right)+c\end{aligned}\)
Answer & Solution
Correct Answer
(B) \(\log (\sin x+\cos x+2)+4 x-4 \sqrt{2} \tan ^{-1}\left(\frac{1+\tan \frac{x}{2}}{\sqrt{2}}\right)+c\)
Step-by-step Solution
Detailed explanation
\(I=\int \frac{3 \sin x+5 \cos x+4}{\sin x+\cos x+2} d x\) Let \((3 \sin x+5 \cos x+4)=A(\sin x+\cos x+2)\) \(+B \frac{d}{d x}(\sin x+\cos x+2)+c\) \(\Rightarrow 3 \sin x+5 \cos x+4\) \(=(A-B) \sin x+(A+B) \cos x+(2 A+C)\) On comparing, we get \(A-B=3, A+B=5\) and \(2 A+C=4\)…
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