Mathematics

If $$\displaystyle f\left ( -x \right )+f\left ( x \right )= 0$$ then $$\displaystyle \int_{0}^{x}f\left ( t \right )$$ dt is


ANSWER

an even function


SOLUTION
Given that f is and odd function.
Let $$g\left( x \right) =\int _{ 0 }^{ x } f\left( t \right) dt$$
Now, $$g\left( -x \right) =\int _{ 0 }^{ -x } f\left( t \right) dt=-\int _{ 0 }^{ x } f\left( -t \right) dt=\int _{ 0 }^{ x } f\left( t \right) dt=g\left( x \right) $$     ...{ Since, $$f$$ is an odd function }
Therefore, $$g$$ is an odd function.
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Single Correct Medium Published on 17th 09, 2020
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