AP EAMCET · Maths · Indefinite Integration
\(\int \frac{\cos x-\sin x}{5+\sin (2 x)} d x=\)
- A \(\frac{1}{2} \cot ^{-1}\left[\frac{1}{2}(\sin x+\cos x)\right]+c\)
- B \(\frac{1}{2} \tan ^{-1}\left[\frac{1}{2}(\sin x+\cos x)\right]+c\)
- C \(\frac{1}{2} \sin ^{-1}\left[\frac{1}{2}(\sin x+\cos x)\right]+c\)
- D \(\frac{1}{2} \cos ^{-1}\left[\frac{1}{2}(\sin x+\cos x)\right]+c\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{2} \tan ^{-1}\left[\frac{1}{2}(\sin x+\cos x)\right]+c\)
Step-by-step Solution
Detailed explanation
\[ \text { } \begin{aligned} I & =\int \frac{\cos x-\sin x}{5+\sin 2 x} d x \\ & =\int \frac{\cos x-\sin x}{4+(\sin x+\cos x)^2} d x \end{aligned} \] Let \(\sin x+\cos x=t \Rightarrow(\cos x-\sin x) d x=d t\)…
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