AP EAMCET · Maths · Definite Integration
If \(\mathrm{m} \in \mathrm{Z}^{+}, \mathrm{n}=2 \mathrm{~m}\) and \(\int_0^{\frac{\pi}{2}} \sin ^{\mathrm{m}} \mathrm{x} \cos ^{\mathrm{n}} \mathrm{x} d \mathrm{x}=\mathrm{K}(\mathrm{m})\) \(\int_0^{\frac{\pi}{2}} \sin ^m x d x\), then \(\frac{2^{m-1}(m-1) !}{(2 m-1) !} K(m)=\)
- A \(\frac{1}{m+2} \frac{1}{m+4} \frac{1}{m+r} \ldots \frac{1}{3 m}\)
- B \(\frac{1}{2 m+2} \frac{1}{2 m+4} \ldots \frac{1}{3 m}\)
- C \(\frac{\pi}{2} \frac{1}{m+2} \frac{1}{m+4} \frac{1}{m+r} \ldots \frac{1}{3 m}\)
- D \(\frac{\pi}{2} \frac{1}{2 m+2} \frac{1}{2 m+4} \ldots \frac{1}{3 m}\)
Answer & Solution
Correct Answer
(A) \(\frac{1}{m+2} \frac{1}{m+4} \frac{1}{m+r} \ldots \frac{1}{3 m}\)
Step-by-step Solution
Detailed explanation
From \(\mathrm{eq}^{\mathrm{n}}\) (i) \(\mathrm{k}(\mathrm{m})=\frac{2 m-1}{3 m} \cdot \frac{2 m-3}{3 m-2} \cdot \frac{2 m-5}{3 m-4} \ldots \frac{3}{m+4} \cdot \frac{1}{m+2}\) Now, \(\frac{2^{m-1}(\mathrm{~m}-1) !}{(2 \mathrm{~m}-1) !} k(\mathrm{~m})\)…
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