Question

Integrate the following with respect to x.

${\displaystyle \frac{{\text{e}}^{3x}-{\text{e}}^{-3x}}{{\text{e}}^x}}$

Answer 1

${\displaystyle \int \left(\frac{{\text{e}}^{3x}-{\text{e}}^{-3x}}{{\text{e}}^x}\right) \text{d}x=\int (\frac{{\text{e}}^{3x}}{{\text{e}}^x}-\frac{{\text{e}}^{-3x}}{{\text{e}}^x}) \text{d}x}$

= ${\displaystyle \int ({\text{e}}^{2x}-{\text{e}}^{-4x}) \text{d}x}$

= ${\displaystyle \frac{{\text{e}}^{2x}}{2}-\frac{{\text{e}}^{-4x}}{-4}+\text{c}}$

= ${\displaystyle \frac{{\text{e}}^{2x}}{2}+\frac{{\text{e}}^{-4x}}{4}+\text{c}}$