If the value of int_{0}^{pi/2} sin^{4}(x) cdo | vedclass

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(A) We are given the property of definite integrals: $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a - x) dx$. Applying this property to the integral $I = \i

Vedclass · Accepted Answer

$\frac{\pi}{32}$

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(A) We are given the property of definite integrals: $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a - x) dx$. Applying this property to the integral $I = \int_{0}^{\pi/2} \cos^{4}(x) \cdot \sin^{2}(x) dx$,we replace $x$ with $(\frac{\pi}{2} - x)$. Since $\cos(\frac{\pi}{2} - x) = \sin(x)$ and $\sin(\frac{\pi}{2} - x) = \cos(x)$,the integral becomes: $I = \int_{0}^{\pi/2} \sin^{4}(x) \cdot \cos^{2}(x) dx$. Given that $\int_{0}^{\pi/2} \sin^{4}(x) \cdot \cos^{2}(x) dx = \frac{\pi}{32}$,it follows that $I = \frac{\pi}{32}$.

If the value of $\int_{0}^{\pi/2} \sin^{4}(x) \cdot \cos^{2}(x) dx = \frac{\pi}{32}$,then the value of $\int_{0}^{\pi/2} \cos^{4}(x) \cdot \sin^{2}(x) dx$ is:

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