int_{-pi / 2}^{pi / 2} sin ^4 x cos ^6 x , dx | vedclass

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(B) Using the property $\int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx$ if $f(x)$ is an even function:Since $\sin^4(-x)\cos^6(-x) = \sin^4 x \co

Vedclass · Accepted Answer

$\frac{3 \pi}{256}$

Vedclass · Answer

(B) Using the property $\int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx$ if $f(x)$ is an even function:Since $\sin^4(-x)\cos^6(-x) = \sin^4 x \cos^6 x$,the function is even.$I = 2 \int_{0}^{\pi / 2} \sin^4 x \cos^6 x \, dx$Using Wallis' Formula: $\int_{0}^{\pi / 2} \sin^m x \cos^n x \, dx = \frac{[(m-1)(m-3)...][(n-1)(n-3)...]}{(m+n)(m+n-2)...} 	imes \frac{\pi}{2}$ (if both $m, n$ are even).Here $m=4, n=6$:$I = 2 	imes \left[ \frac{(3 \cdot 1) 	imes (5 \cdot 3 \cdot 1)}{(10 \cdot 8 \cdot 6 \cdot 4 \cdot 2)} 	imes \frac{\pi}{2} ight]$$I = 2 	imes \left[ \frac{3 	imes 15}{3840} 	imes \frac{\pi}{2} ight] = \frac{45 \pi}{3840} = \frac{3 \pi}{256}$.

$\int_{-\pi / 2}^{\pi / 2} \sin ^4 x \cos ^6 x \, dx$ is equal to

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