The value of the integral int_{-pi}^{pi} sin( | vedclass

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(A) Let $I = \int_{-\pi}^{\pi} \sin(mx) \sin(nx) \, dx$.Since the integrand $f(x) = \sin(mx) \sin(nx)$ is an even function,we can write:$I = 2 \int_{0

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(A) Let $I = \int_{-\pi}^{\pi} \sin(mx) \sin(nx) \, dx$.Since the integrand $f(x) = \sin(mx) \sin(nx)$ is an even function,we can write:$I = 2 \int_{0}^{\pi} \sin(mx) \sin(nx) \, dx$.Using the trigonometric identity $2 \sin A \sin B = \cos(A - B) - \cos(A + B)$,we get:$I = \int_{0}^{\pi} [\cos((m - n)x) - \cos((m + n)x)] \, dx$.Integrating term by term:$I = \left[ \frac{\sin((m - n)x)}{m - n} - \frac{\sin((m + n)x)}{m + n} \right]_{0}^{\pi}$.Evaluating at the limits:$I = \left( \frac{\sin((m - n)\pi)}{m - n} - \frac{\sin((m + n)\pi)}{m + n} \right) - (0 - 0)$.Since $m, n \in I$ (integers),$\sin(k\pi) = 0$ for any integer $k$.Therefore,$I = 0 - 0 = 0$.

The value of the integral $\int_{-\pi}^{\pi} \sin(mx) \sin(nx) \, dx$ for $m \neq n$ $(m, n \in I)$ is:

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