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AP EAPCET 07-Jul-2022 Shift 1 Paper

Section: Mathematics

Question : 76 of 160

Marks: +1, -0

\int\limits_{0}^{\frac{\pi}{2}} \sin^{m} x \cos^{4} x \, dx = \frac{7\pi}{2048} \Rightarrow m=

Solution: 👈: Video Solution

\int\limits_{0}^{\frac{\pi}{2}} \sin^{m} x \cdot \cos^{4} x \, dx = \frac{7\pi}{2048}

\because \int\limits_{0}^{\frac{\pi}{2}} \sin^{n} x \cdot \cos^{m} x \, dx

[(n-1)(n-3)(n-5) \dots 1 \;\text{ or }\; 2] x

= \frac{[(m-1)(m-3)(m-5) \dots 1 \;\text{ or }\; 2]}{(m+n)(m+n-2) \dots 1 \;\text{ or }\; 2} \times k

Where,

k = \frac{\pi}{2}. \text{Both } m \text{ and } n \text{ are even}

I, \text{otherwise}

I = \int\limits_{0}^{\frac{\pi}{2}} \sin^{m} x \cos^{n} x \, dx

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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