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

To evaluate the definite integral of the function \( \frac{e^{x+1}}{e^x} \) from 0 to 1, let's first simplify the integrand:

\[
\frac{e^{x+1}}{e^x} = e^{x+1 - x} = e^1 = e
\]

The integrand simplifies to a constant \( e \). Now, we can compute the definite integral:

\[
\int_0^1 e \, dx
\]

Since \( e \) is a constant, we can take it outside the integral:

\[
e \int_0^1 dx
\]

The integral of \( dx \) from 0 to 1 is:

\[
\int_0^1 dx = [x]_0^1 = 1 - 0 = 1
\]

So, the definite integral becomes:

\[
e \times 1 = e
\]

### Answer:
\[
\int_0^1 \frac{e^{x+1}}{e^x} \, dx = e
\]

Would you like more details or have any questions?

### Related Questions:
1. What is the integral of \( e^x \) from 0 to 1?
2. How would you evaluate the integral of \( e^{2x} \) from 0 to 1?
3. What is the integral of \( \frac{e^x}{x} \) from 1 to \( e \)?
4. How do you compute the integral of a constant?
5. How to evaluate definite integrals with different limits?

### Tip:
When integrating exponential functions, always look for ways to simplify the exponent or apply properties of exponents to make the integration straightforward.

Evaluate the definite integral from 0 to 1 of e^(x+1) / e^x.

Learn how to evaluate the definite integral of e^(x+1) / e^x from 0 to 1. This problem simplifies using properties of exponents and covers key concepts in integral calculus. Suitable for students in Grades 11-12 or early college.

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