Tìm nguyên hàm: \(I = \smallint \frac{{{e^x}dx}}{{{e^x} + 4{e^{ – x}}}}\)

Xét \(J = \smallint \frac{{{e^x}}}{{{e^x} + 4{e^{ – x}}}}\)

Ta xét hệ : 

\(\begin{array}{l}
\left\{ \begin{array}{l}
I + 4J = \int {\frac{{{e^x} + 4{e^{ – x}}}}{{{e^x} + 4{e^{ – x}}}}} dx = \int {dx}  = x + {C_1}\\
I – 4J = \int {\frac{{{e^x} – 4{e^{ – x}}}}{{{e^x} + 4{e^{ – x}}}}} dx = \ln \left| {{e^x} + 4{e^{ – x}}} \right| + {C_2}
\end{array} \right.\\
 \Rightarrow 2I = x + \ln \left| {{e^x} + 4{e^{ – x}}} \right| + {C_1} + {C_2}
\end{array}\)

Hay \(I=\frac{1}{2}x + \frac{1}{2}\ln \left| {{e^x} + 4{e^{ – x}}} \right| + C\).