\( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos x}{e^x+1}dx \)

\( X
= \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos x}{e^x+1}dx \\
= -\int_{\frac{\pi}{2}}^{-{\frac{\pi}{2}}} \frac{\cos( -u)}{e^{-u} + 1}du \\
= \int_{-{\frac{\pi}{2}}}^{\frac{\pi}{2}} \frac{\cos( -u)}{e^{-u} + 1}du \\
= \int_{-{\frac{\pi}{2}}}^{\frac{\pi}{2}} \frac{\cos u}{e^{-u} + 1}du \\
= \int_{-{\frac{\pi}{2}}}^{\frac{\pi}{2}} \frac{\cos x}{e^{-x} + 1}dx 
\)
\(
X + X = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\cos x}{e^x+1}dx + \int_{-{\frac{\pi}{2}}}^{\frac{\pi}{2}} \frac{\cos x}{e^{-x} + 1}dx
\)

\(
\frac{\cos x}{e^x+1} + \frac{\cos x}{e^{-x} + 1} = \frac{\cos x}{e^x+1} + \frac{e^x\cos x}{1+e^x} = \frac{\cos x (e^x + 1)}{e^x + 1} = \cos x
\)

\(
2X = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x dx = 2 \\
X=1
\)