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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(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:

Similar Questions

The value of the integral $\sum\limits_{k = 1}^n {\int_0^1 {f(k - 1 + x)\,dx} } $ is

$\int_{0}^{\frac{\pi}{2}} \frac{\sin^{\frac{2}{3}} x}{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x} dx =$

By using the properties of definite integrals,evaluate the integral $\int_{0}^{4}|x-1| d x$.

$\int_0^\pi \frac{x \, dx}{4 \cos^2 x + 9 \sin^2 x} = $

$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ is equal to . . . . . . .

Vedclass Test Series

Exam Paper Generator

Online Exam Module