Is the Integral of f(t)*cost dt an Odd Function with Limits from -pi/2 to pi/2?

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SUMMARY

The integral of f(t)cos(t) from -π/2 to π/2 is not an odd function; rather, the function y(t) = f(t)cos(t) is classified as odd. The piecewise function f(t) is defined as 1 for -π/2 ≤ t ≤ 0, -1 for 0 ≤ t ≤ π/2, and 0 elsewhere. The relationship y(-t) = -f(t)cos(t) confirms that y(t) is odd, as it satisfies the condition for odd functions. Thus, the product of the even function cos(t) and the odd function f(t) results in an overall odd function.

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f(t) = 1 if -pi/2 <=t <=0
-1 if 0<=t<= pi/2
0 elsewhere

how does integral of f(t)*cost dt become and odd function with the integral limit from - pi/2 to pi/2 ?

thanks a lot
 
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skan said:
how does integral of f(t)*cost dt become and odd function with the integral limit from - pi/2 to pi/2 ?

It's not the integral of f(t)cos(t) that is odd (in fact it's just a number). It's the function f(t)cos(t) itself that is odd.

Let's see why.

Let y(t)=f(t)cos(t)
so, y(-t)=f(-t)cos(-t)

Note that f(-t)=-f(t) and cos(-t)=cos(t). So,

y(-t)=-f(t)cos(t),

and thus y(t) is odd. In general, the product of any even function and any odd function is odd.
 
thanks a lot
 

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