Question

Evaluate each of the following integral:

\[\int_{- a}^a \frac{1}{1 + a^x}dx\]`, a > 0`

Answer 1

\[\text{Let I} = \int_{- a}^a \frac{1}{1 + a^x}dx................\left(1\right)\]

\[I = \int_{- a}^a \frac{1}{1 + a^\left[ a + \left( - a \right) - x \right]}dx\]
\[ = \int_{- a}^a \frac{1}{1 + a^{- x}}dx\]
\[ = \int_{- a}^a \frac{a^x}{a^x + 1}dx ..................\left( 2 \right)\]

Adding (1) and (2), we get

\[2I = \int_{- a}^a \frac{1 + a^x}{1 + a^x}dx\]
\[ \Rightarrow 2I = \int_{- a}^a dx\]
\[ \Rightarrow 2I = \left.x\right|_{- a}^a \]
\[ \Rightarrow 2I = a - \left( - a \right) = 2a\]
\[ \Rightarrow I = a\]