Question

If f(x) is a continuous function defined on [−a, a], then prove that 

\[\int\limits_{- a}^a f\left( x \right) dx = \int\limits_0^a \left\{ f\left( x \right) + f\left( - x \right) \right\} dx\]

Answer 1

\[Let\ I = \int_{- a}^a f\left( x \right) d x\]
\[\text{By Additive property}\]
\[I = \int_{- a}^0 f\left( x \right) d x + \int_0^a f\left( x \right) d x\]
\[Let x = - t, then\ dx = - dt, \]
\[When\ x = - a, t = a, x = 0, t = 0\]
\[Hence\ \int_{- a}^0 f\left( x \right) d x = - \int_a^0 f\left( - t \right) d t\]
\[ = \int_0^a f\left( - t \right) d t = \int_0^a f\left( - x \right) dx .......................\left( \text{Changing the variable} \right)\]
Therefore,
\[I = \int_0^a f\left( - x \right) d x + \int_0^a f\left( x \right) d x\]
\[ = \int_0^a \left\{ f\left( x \right) + f\left( - x \right) \right\} dx\]
\[\text{Hence, proved} .\]