The value of int_{-pi/2}^{pi/2} (3sin x + sin | vedclass

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Question

(C) Let $I = \int_{-\pi/2}^{\pi/2} (3\sin x + \sin^3 x) \, dx$.Consider the function $f(x) = 3\sin x + \sin^3 x$.Check if the function is even or odd

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(C) Let $I = \int_{-\pi/2}^{\pi/2} (3\sin x + \sin^3 x) \, dx$.Consider the function $f(x) = 3\sin x + \sin^3 x$.Check if the function is even or odd by substituting $-x$ for $x$:$f(-x) = 3\sin(-x) + \sin^3(-x)$Since $\sin(-x) = -\sin x$,we have:$f(-x) = 3(-\sin x) + (-\sin x)^3 = -3\sin x - \sin^3 x = -(3\sin x + \sin^3 x) = -f(x)$.Since $f(-x) = -f(x)$,the function $f(x)$ is an odd function.According to the property of definite integrals,if $f(x)$ is an odd function,then $\int_{-a}^{a} f(x) \, dx = 0$.Therefore,$\int_{-\pi/2}^{\pi/2} (3\sin x + \sin^3 x) \, dx = 0$.

The value of $\int_{-\pi/2}^{\pi/2} (3\sin x + \sin^3 x) \, dx$ is

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