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

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Answer 1 · 2014-09-21T19:16:41

Let #f(x)=sin(mx)cos(nx)#.

Let us consider

#f(-x)=sin(-mx)cos(-nx)#

since #sin(mx)# is odd, and #cos(nx)# is even,

#=-sin(mx)cos(nx)=-f(x)#,

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

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

#int_{-a}^a f(x)dx=0#.

Hence,

#int_{-pi}^{pi}sin(mx)cos(nx) dx=0#