Problem 01 Area between curves

Find the area of the region described. The region bounded by

$$y=e^x\text{ and }y=e^{-2x}\text{, and }x=\ln (5)$$

This is not a very difficult problem. The reason why I added this to the list is I confused x = ln(5) with y = ln(5), then I came up with so many possible solutions. First, I will solve the original problem, and then solve my “made up” problem with various ways.

The are of the region made by the two functions and vertical line x = ln(5) can be calculated with the integration expression:

$${\int }_0^{\ln (5)}(e^x-e^{-2x})dx$$ $$=\left(e^x+\frac{e^{-2x}}{2}\right){|}_0^{\ln (5)}$$ $$=\left(5+\frac{1}{50}\right)-\left(1+\frac{1}{2}\right)=\frac{88}{25}$$

very easy huh? Now let’s modify the problem into my delulu:

$$y=e^x\text{ and }y=e^{-2x}\text{, and }y=\ln (5)$$

To solve the modified version, there are two methods: 1.