Question

If f is an integrable function, show that

\[\int\limits_{- a}^a x f\left( x^2 \right) dx = 0\]

 

Answer 1

\[I = \int_{- a}^a xf\left( x^2 \right) d x\]
\[Let\ g\left( x \right) = xf\left( x^2 \right)\]
\[ \Rightarrow g\left( - x \right) = \left( - x \right)f \left( - x \right)^2 = - \left( x \right)f\left( x^2 \right) = - g\left( x \right) i . e, g\left( x \right) \text{ is odd }\]
Therefore
\[I = 0 ...............\left[\text{Using }\int_{- a}^a g\left( x \right) d x\ = 0\text{ when g(x) is odd}\right]\]