MHT CET · Maths · Definite Integration
\( \int_0^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1-\sin x \cos x} d x= \)
- A \(\frac{\pi}{4}\)
- B \(\frac{2}{\pi}\)
- C 0
- D \(\frac{\pi}{2}\)
Answer & Solution
Correct Answer
(C) 0
Step-by-step Solution
Detailed explanation
Let \(I=\int_0^{\pi / 2} \frac{\sin x-\cos x}{1-\sin x \cos x} d x\)
\( \therefore I=\int_0^{\frac{\pi}{2}} \frac{\sin \left(\frac{\pi}{2}-x\right)-\cos \left(\frac{\pi}{2}-x\right)}{1-\sin \left(\frac{\pi}{2}-x\right) \cos \left(\frac{\pi}{2}-x\right)} d x \)
\( =\int_0^{\pi / 2} \frac{\cos x-\sin x}{1-\cos x \sin x} d x\)
Eq. (1) \(+(2)\) gives
\(
2 \mathrm{I}=\int_0^{\frac{\pi}{2}} 0 \mathrm{dx} \Rightarrow \mathrm{I}=0
\)
\( \therefore I=\int_0^{\frac{\pi}{2}} \frac{\sin \left(\frac{\pi}{2}-x\right)-\cos \left(\frac{\pi}{2}-x\right)}{1-\sin \left(\frac{\pi}{2}-x\right) \cos \left(\frac{\pi}{2}-x\right)} d x \)
\( =\int_0^{\pi / 2} \frac{\cos x-\sin x}{1-\cos x \sin x} d x\)
Eq. (1) \(+(2)\) gives
\(
2 \mathrm{I}=\int_0^{\frac{\pi}{2}} 0 \mathrm{dx} \Rightarrow \mathrm{I}=0
\)
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