Mathematics

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

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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