If the value of the integral _-1^1 x1+3^xdx is (2)/( ).Then, a value of is

Question

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

Accepted Answer

$\begin{gathered} \int _{-1}^{1}\frac{\cos \alpha x}{1+{3}^{x}}dx=\frac{2}{\pi } \ I =\int _{-1}^{1}\frac{\cos \alpha x}{1+{3}^{x}}dx \ \Rightarrow I =\int _{0}^{1}(\frac{\cos \alpha x}{1+{3}^{x}}+\frac{\cos \alpha x}{1+{3}^{-x}})dx \ =\int _{0}^{1}\cos \alpha xdx \ ={(\frac{\sin \alpha x}{\alpha })}_{1}^{0} \ =\frac{\sin \alpha }{\alpha } \ \Rightarrow \frac{\sin \alpha }{\alpha }=\frac{2}{\pi } \ \Rightarrow \alpha =\frac{\pi }{2} \end{gathered}$

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