Question

Why is it very easy to conclude that${\int }_0^1\frac{1}{e^x}dx$ converges without making any integration calculations?

Short answer

The integral $\frac{1}{e^x}$is continuous on the interval $\left[0,1\right]$. The function $\frac{1}{e^x}$has no vertical asymptotes, so this is not improper integral. So, it is easy to conclude that given integral converges without making any integration calculations.

Step-by-step solution

Step 1. Given information

${\int }_0^1\frac{1}{e^x}dx$.

Step 2. The given integral converges.

The integral$\frac{1}{e^x}$is continuous on the interval$\left[0,1\right]$.

The function $\frac{1}{e^x}$has no vertical asymptotes, so this is not improper integral.

From the above information, it is easy to conclude that given integral converges without making any integration calculations.

Step 3. The graph of y=1ex is shown below: