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Question: Answered & Verified by Expert
$\int_0^{\pi / 2} \sin ^m x \cos ^4 x d x=\frac{7 \pi}{2048} \Rightarrow m=$
MathematicsDefinite IntegrationAP EAMCETAP EAMCET 2022 (07 Jul Shift 1)
Options:
  • A $8$
  • B $6$
  • C $10$
  • D $12$
Solution:
2852 Upvotes Verified Answer
The correct answer is: $8$
$\int_0^{\pi / 2} \sin ^m x \cdot \cos ^4 x d x=\frac{7 \pi}{2048}$


where, $k=\left\{\begin{array}{l}\frac{\pi}{2}, \text { Both } m \text { and } n \text { are even } \\ 1, \text { otherwise }\end{array}\right.$
$I=\int_0^{\pi / 2} \sin ^m x \cos ^n x d x$
If $m=8, n=4$, then
$I=\left[\frac{(7 \cdot 5 \cdot 3 \cdot 1)(3 \cdot 1)}{12 \cdot 10 \cdot 8 \cdot 6 \cdot 4 \cdot 2}\right] \cdot \frac{\pi}{2}=\frac{7 \pi}{2048}$
$\therefore m=8$

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