evalutate dfinite integral from 0 to 1 of e^(x+1)   /  e^x

We need to evaluate the definite integral:

\[
\int_0^1 \frac{e^{x+1}}{e^x} \, dx
\]

### Step-by-step solution:

1. Simplify the integrand:
   \[
   \frac{e^{x+1}}{e^x} = \frac{e^x \cdot e^1}{e^x} = e^1 = e
   \]

   So the integral simplifies to:
   \[
   \int_0^1 e \, dx
   \]

2. Now, we can factor out the constant \(e\):
   \[
   e \int_0^1 1 \, dx
   \]

3. The integral of \(1\) over the interval \([0, 1]\) is simply the length of the interval, which is \(1 - 0 = 1\):
   \[
   e \cdot 1 = e
   \]

Thus, the value of the definite integral is:

\[
e
\]

Would you like more details on this? Here are 5 related questions you might find interesting:

1. How would the solution change if the limits of integration were different?
2. What is the interpretation of this integral geometrically?
3. How can you compute a similar integral for more complex functions?
4. How would you approach integrals involving trigonometric functions in the numerator?
5. Can we apply substitution or integration by parts to this problem?

**Tip:** Always simplify the integrand first to make the integral easier to solve!

evaluate definite integral from 0 to 1 of e^(x+1) / e^x

Learn how to evaluate the definite integral from 0 to 1 of e^(x+1) / e^x using simplification techniques. This problem introduces key concepts from calculus, such as definite integrals and exponential functions. Suitable for students in Grades 11-12.

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