MHT CET · Maths · Indefinite Integration
\(\int \frac{\sin x}{\sin \left(x-\frac{\pi}{4}\right)} d x=\)
- A \(\frac{1}{\sqrt{2}}\left[x+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|\right]+c\)
- B \(x+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c\)
- C \(x-\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c\)
- D \(\frac{1}{\sqrt{2}}\left[x-\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|\right]+c\)
Answer & Solution
Correct Answer
(A) \(\frac{1}{\sqrt{2}}\left[x+\log \left|\sin \left(x-\frac{\pi}{4}\right)\right|\right]+c\)
Step-by-step Solution
Detailed explanation
\(I =\int \frac{\sin x}{\sin \left(x-\frac{\pi}{4}\right)} d x \)
\( =\int \frac{\sin \left(x-\frac{\pi}{4}+\frac{\pi}{4}\right)}{\sin \left(x-\frac{\pi}{4}\right)} d x=\int \frac{\sin \left(x-\frac{\pi}{4}\right) \cos \frac{\pi}{4}+\cos \left(x-\frac{\pi}{4}\right) \sin \frac{\pi}{4}}{\sin \left(x-\frac{\pi}{4}\right)} \)
\( =\int\left(\frac{1}{\sqrt{2}}+\cot \left(x-\frac{\pi}{4}\right) \cdot \frac{1}{\sqrt{2}}\right) d x=\int \frac{1}{\sqrt{2}}\left[1+\cot \left(x-\frac{\pi}{4}\right)\right] d x \)
\( =\frac{1}{\sqrt{2}}\left[\int 1 d x+\int \cot \left(x-\frac{\pi}{4}\right)\right] d x=\frac{1}{\sqrt{2}}\left[x+\log \left(\sin \left(x-\frac{\pi}{4}\right)\right]+c\right.\)
\( =\int \frac{\sin \left(x-\frac{\pi}{4}+\frac{\pi}{4}\right)}{\sin \left(x-\frac{\pi}{4}\right)} d x=\int \frac{\sin \left(x-\frac{\pi}{4}\right) \cos \frac{\pi}{4}+\cos \left(x-\frac{\pi}{4}\right) \sin \frac{\pi}{4}}{\sin \left(x-\frac{\pi}{4}\right)} \)
\( =\int\left(\frac{1}{\sqrt{2}}+\cot \left(x-\frac{\pi}{4}\right) \cdot \frac{1}{\sqrt{2}}\right) d x=\int \frac{1}{\sqrt{2}}\left[1+\cot \left(x-\frac{\pi}{4}\right)\right] d x \)
\( =\frac{1}{\sqrt{2}}\left[\int 1 d x+\int \cot \left(x-\frac{\pi}{4}\right)\right] d x=\frac{1}{\sqrt{2}}\left[x+\log \left(\sin \left(x-\frac{\pi}{4}\right)\right]+c\right.\)
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