Evaluate the definite integral int_{0}^{frac{ | vedclass

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(B) Let $I = \int_{0}^{\frac{\pi}{2}} \cos ^{2} x \,d x$. Using the trigonometric identity $\cos ^{2} x = \frac{1 + \cos 2x}{2}$,we have: $I = \int_{0

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$\frac{\pi}{4}$

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(B) Let $I = \int_{0}^{\frac{\pi}{2}} \cos ^{2} x \,d x$. Using the trigonometric identity $\cos ^{2} x = \frac{1 + \cos 2x}{2}$,we have: $I = \int_{0}^{\frac{\pi}{2}} \frac{1 + \cos 2x}{2} \,d x$ $I = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} (1 + \cos 2x) \,d x$ $I = \frac{1}{2} \left[ x + \frac{\sin 2x}{2} \right]_{0}^{\frac{\pi}{2}}$ Now,applying the limits: $I = \frac{1}{2} \left[ \left( \frac{\pi}{2} + \frac{\sin \pi}{2} \right) - \left( 0 + \frac{\sin 0}{2} \right) \right]$ Since $\sin \pi = 0$ and $\sin 0 = 0$: $I = \frac{1}{2} \left[ \frac{\pi}{2} + 0 - 0 - 0 \right] = \frac{\pi}{4}$.

Evaluate the definite integral $\int_{0}^{\frac{\pi}{2}} \cos ^{2} x \,d x$.

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