Lời giải

Đặt \( x = -t \)

\( I = \int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{1}{(1 + e^x)(1 - x^2)} \, dx = \int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{1}{(1 + e^{-t})(1 - t^2)} \, dt\)

\( I = \int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{e^x}{(1 + e^x)(1 - x^2)} \, dx = \int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{e^x + 1 - 1}{(1 + e^x)(1 - x^2)} \, dx \)

\( I = \int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{1}{1 - x^2} \, dx - I \)

\(\Rightarrow I = \frac{1}{2} \int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{1}{1 - x^2} \, dx = \frac{1}{2} \ln 3\)