If the value of the integral [_-1^1 ( x)/(1+3^x) d x] is [(2)/()].Then, a value of [] is

Question

If the value of the integral $$\int\limits_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x$$ is $$\frac{2}{\pi}$$.Then, a value of $$\alpha$$ is

Accepted Answer

$$\begin{aligned}
& \text { Given, } \int_\limits{-1}^1 \frac{\cos \alpha x}{1+3^x} d x=\frac{2}{\pi} \
& \begin{aligned}
I & =\int_\limits{-1}^1 \frac{\cos \alpha x}{1+3^x} d x \
\Rightarrow I & =\int_\limits0^1\left(\frac{\cos \alpha x}{1+3^x}+\frac{\cos \alpha x}{1+3^{-x}}\right) d x \
& =\int_\limits0^1 \cos \alpha x d x \
& =\left(\frac{\sin \alpha x}{\alpha}\right)_0^1 \
& =\frac{\sin \alpha}{\alpha} \
\Rightarrow & \frac{\sin \alpha}{\alpha}=\frac{2}{\pi} \
\Rightarrow & \alpha=\frac{\pi}{2}
\end{aligned}
\end{aligned}$$

Solutions · 1