Question

Evaluate each of the following integrals:

${\int }_{-a}^a\frac{1}{1+a^x}dx$ , a > 0

Answer 1

Let I = ${\int }_{-a}^a\frac{1}{1+a^x}dx$ .....(1)
Then,
$$\begin{gathered} I={\int }_{-a}^a\frac{1}{1+a^{\left[a+\left(-a\right)-x\right]}}dx \\ ={\int }_{-a}^a\frac{1}{1+a^{-x}}dx \\ ={\int }_{-a}^a\frac{a^x}{a^x+1}dx.....\left(2\right) \end{gathered}$$

Adding (1) and (2), we get

$$\begin{gathered} 2I={\int }_{-a}^a\frac{1+a^x}{1+a^x}dx \\ \Rightarrow 2I={\int }_{-a}^{a}dx \\ \Rightarrow 2I={\overline{)x}}_{-a}^a \\ \Rightarrow 2I=a-\left(-a\right)=2a \\ \Rightarrow I=a \end{gathered}$$