What is the value of int limits_0^1 cos (pi x | vedclass

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(C) Let $I = \int \limits_0^1 \cos (\pi x) \cos ([2 x] \pi) d x$.Since $[2x]$ is a step function,we split the integral at $x = \frac{1}{2}$:For $0 \le

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$\frac{2}{\pi}$

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(C) Let $I = \int \limits_0^1 \cos (\pi x) \cos ([2 x] \pi) d x$.Since $[2x]$ is a step function,we split the integral at $x = \frac{1}{2}$:For $0 \le x For $\frac{1}{2} \le x Thus,$I = \int \limits_0^{1/2} \cos(\pi x) \cdot (1) d x + \int \limits_{1/2}^1 \cos(\pi x) \cdot (-1) d x$.$I = \left[ \frac{\sin(\pi x)}{\pi} \right]_0^{1/2} - \left[ \frac{\sin(\pi x)}{\pi} \right]_{1/2}^1$.$I = \frac{1}{\pi} [\sin(\frac{\pi}{2}) - \sin(0)] - \frac{1}{\pi} [\sin(\pi) - \sin(\frac{\pi}{2})]$.$I = \frac{1}{\pi} [1 - 0] - \frac{1}{\pi} [0 - 1] = \frac{1}{\pi} + \frac{1}{\pi} = \frac{2}{\pi}$.

What is the value of $\int \limits_0^1 \cos (\pi x) \cos ([2 x] \pi) d x$? (Here $[t]$ denotes the greatest integer function of the real number $t$.)

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