If f is an integrable function, show that: ^ a _ -a xf (x^2 )dx=0 - Vidyadip

Question

If f is an integrable function, show that:$\int\limits^{\text{a}}_{-\text{a}}\text{xf}\big(\text{x}^2\big)\text{dx}=0$

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Answer

$\text{I}=\int\limits^{\text{a}}_{-\text{a}}\text{xf}(\text{x}^2)\text{dx}$Let $\text{g(x)}=\text{f}\big(\text{x}^2\big)$
$\Rightarrow\text{g}(-\text{x})=(-\text{x})\text{f}(-\text{x})^2=-(\text{x})\text{f}(\text{x})^2=-\text{g(x)}\text{ i.e., g(x) is even}$
Therefore,
$\text{I}=0$ $\bigg[\text{Using}\int\limits^{\text{a}}_{-\text{a}}\text{g}(\text{x})\text{dx}=0\text{ when g(x) is odd}\bigg]$

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