Can you find the integral of e^sqrt(x-1) / sqrt(x-1) dx

With upper limit: infinity, lower limit: 1

To evaluate the integral 

\[
\int_1^{\infty} \frac{e^{\sqrt{x-1}}}{\sqrt{x-1}} \, dx,
\]

we can use a substitution. Let:

\[
u = \sqrt{x-1} \implies x = u^2 + 1 \implies dx = 2u \, du.
\]

The limits change as follows:

- When \( x = 1 \), \( u = 0 \).
- When \( x \to \infty \), \( u \to \infty \).

Now, substituting these into the integral:

\[
\int_0^{\infty} \frac{e^u}{u} (2u) \, du = 2 \int_0^{\infty} e^u \, du.
\]

This simplifies to:

\[
2 \int_0^{\infty} e^u \, du.
\]

However, the integral 

\[
\int_0^{\infty} e^u \, du
\]

diverges, because \( e^u \) grows exponentially as \( u \to \infty \). Thus, the original integral also diverges.

Therefore, the integral 

\[
\int_1^{\infty} \frac{e^{\sqrt{x-1}}}{\sqrt{x-1}} \, dx
\]

diverges.

### Summary
- The integral diverges.

Would you like further details or explanations on any part of this process? Here are some questions for further exploration:

1. How do substitution methods work in integrals?
2. What are the implications of an integral diverging?
3. Can you provide examples of convergent integrals involving exponential functions?
4. What other techniques can be used to evaluate improper integrals?
5. How does the behavior of \( e^{\sqrt{x-1}} \) impact convergence? 

**Tip:** Always check the behavior of the integrand as it approaches the limits of integration to determine convergence or divergence.

Can you find the integral of e^sqrt(x-1) / sqrt(x-1) dx with upper limit: infinity, lower limit: 1

Explore the step-by-step solution to evaluating the improper integral of e^sqrt(x-1) / sqrt(x-1) with limits from 1 to infinity. This solution uses substitution and examines the conditions for convergence or divergence, explaining why the integral diverges. Ideal for students in undergraduate calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation