Giải thích các bước giải:
\(I=\int \frac{e^{x}}{e^{x}+e^{-x}}dx\)
\( \Leftrightarrow I=\int \frac{e^{x}}{e^{x}+\frac{1}{e^{x}}}dx\)
\( \Leftrightarrow \int \frac{e^{x}}{\frac{e^{2x}+1}{e^{x}}}dx\)
\(I=\int \frac{e^{2x}}{e^{2x}+1}dx\)
Đặt \(t=e^{2x}+1\)
\(dt=2.e^{2x}.dx \Leftrightarrow \frac{dt}{2}=e^{2x}.dx\)
\(\rightarrow I=\int \frac{\frac{dt}{2}}{t}=\int \frac{1}{2t}dt=\frac{1}{2}.\int \frac{1}{t}.dt=\frac{1}{2}.ln(|t|)=\frac{1}{2}ln(|e^{2x}+1|)+C=\frac{1}{2}ln(e^{2x}+1)+C\)