How do you prove the given formula where m and n are positive integers? $\int_{- π}^{π} sin (m x) \cdot cos (n x) d x = 0$ $int_-pi^pisin(mx)*cos(nx)dx = 0$

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Answer 1 · 2014-09-21T19:16:41

Let $f (x) = sin (m x) cos (n x)$ $f(x)=sin(mx)cos(nx)$ .

Let us consider

$f (- x) = sin (- m x) cos (- n x)$ $f(-x)=sin(-mx)cos(-nx)$

since $sin (m x)$ $sin(mx)$ is odd, and $cos (n x)$ $cos(nx)$ is even,

$= - sin (m x) cos (n x) = - f (x)$ $=-sin(mx)cos(nx)=-f(x)$ ,

which means that $f (x)$ $f(x)$ is an odd function.

Recall that if $f (x)$ $f(x)$ is odd,

$\int_{- a}^{a} f (x) d x = 0$ $int_{-a}^a f(x)dx=0$ .

Hence,

$\int_{- π}^{π} sin (m x) cos (n x) d x = 0$ $int_{-pi}^{pi}sin(mx)cos(nx) dx=0$

How do you prove the given formula where m and n are positive integers? ∫π−πsin(mx)⋅cos(nx)dx=0int_-pi^pisin(mx)*cos(nx)dx = 0