intlimits_0^{frac{pi }{4}} {sin 2x} ,dx is eq | vedclass

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(C) We need to evaluate the definite integral $I = \int\limits_0^{\frac{\pi }{4}} \sin 2x \,dx$.Using the integration formula $\int \sin(ax) \,dx = -\

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$0.5$

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(C) We need to evaluate the definite integral $I = \int\limits_0^{\frac{\pi }{4}} \sin 2x \,dx$.Using the integration formula $\int \sin(ax) \,dx = -\frac{1}{a} \cos(ax) + C$,we get:$I = \left[ -\frac{\cos 2x}{2} \right]_0^{\frac{\pi }{4}}$Applying the limits:$I = -\frac{1}{2} \left[ \cos(2 \cdot \frac{\pi }{4}) - \cos(2 \cdot 0) \right]$$I = -\frac{1}{2} \left[ \cos(\frac{\pi }{2}) - \cos(0) \right]$Since $\cos(\frac{\pi }{2}) = 0$ and $\cos(0) = 1$:$I = -\frac{1}{2} [0 - 1] = -\frac{1}{2} (-1) = 0.5$.

$\int\limits_0^{\frac{\pi }{4}} {\sin 2x} \,dx$ is equal to

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