MHT CET · Maths · Definite Integration
\(\frac{\pi}{2}\)
If \(\int_0 \frac{\cot x}{\cot x+\operatorname{cosec} x} d x=m(\pi+n)\), then (m.n) equals
- A \(\frac{1}{2}\)
- B \(-1\)
- C \(-\frac{1}{2}\)
- D 1
Answer & Solution
Correct Answer
(B) \(-1\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \int_0^{\frac{\pi}{2}} \frac{\cot x}{\cot x+\operatorname{cosec} x} d x=\int_0^{\frac{\pi}{2}} \frac{\cos x}{\cos x+1} d x \\ & =\int_0^{\frac{\pi}{2}} \frac{2 \cos ^2 \frac{x}{2}-1}{2 \cos ^2 \frac{x}{2}} d x=\int_0^{\frac{\pi}{2}}\left(1-\frac{1}{2} \sec ^2 \frac{x}{2}\right) d x \\ & =\left[x-\tan \frac{x}{2}\right]_0^{\pi / 2}=\frac{\pi}{2}-1=\frac{1}{2}(\pi+(-2)) \\ & \Rightarrow m=\frac{1}{2} \text { and } n=-2 \\ & \Rightarrow \mathrm{m} \times \mathrm{n}=\frac{1}{2} \times(-2)=-1\end{aligned}\)
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