Tìm nguyên hàm của hàm số sau: ∫dx/ex−e−x

Question

\(\int \frac{{\rm d}x}{{\rm e}^{x} -{\rm e}^{-x} }\)

Answer 1

\(\int \frac{{\rm d}x}{{\rm e}^{x} -{\rm e}^{-x} } =\int \frac{{\rm e}^{x} {\rm d}x}{{\rm e}^{2x} -1} \)

Đặt \(t={\rm e}^{x} \Rightarrow {\rm d}t={\rm e}^{x} {\rm d}x\)

Khi đó: \(\int \frac{{\rm e}^{x} {\rm d}x}{{\rm e}^{2x} -1} =\int \frac{{\rm d}t}{t^{2} -1} =\int \frac{{\rm d}t}{\left(t-1\right)\left(t+1\right)} =\frac{1}{2} \ln \left|\frac{t-1}{t+1} \right|+C=\frac{1}{2} \ln \left|\frac{{\rm e}^{x} -1}{{\rm e}^{x} +1} \right|+C.\)