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Question

If f(t) is an odd function, then 0xf(t)dt is


A

An odd function

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B

An even function

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C

Neither even nor odd

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D

0

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Solution

The correct option is B

An even function


Explanation for the correct option:

Step 1: Determine the function

Given function f(t) is an odd function

We can write

-f(t)=f(-t)

Let g-x=0-xf(t)dt and t=-y,dt=-dy

Step 2: Applying limits,

t=0y=0t=-xy=x

g(-x)=0xf(-y)(-dy)=-x0f(y)dy+0xf(y)dy=0+0xf(y)dyg(-x)=0-xf(t)dt,wecansayg(x)=0xf(t)dt=g(x)g(-x)=g(x)

Therefore, 0xf(t)dt is an even function.

Hence, the correct option is an option (B).


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