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Chapter 12: Problem 19

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x $$

Short Answer

Expert verified
The given integral \[ \int_{1}^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}} dx \] diverges and when evaluated, yields an infinite result.

Step by step solution

01

Check for Convergence or Divergence

An improper integral diverges if the limit of the definite integral from a regular number a to a point where function becomes infinite (either a positive or negative infinity) is infinity. If this limit is a real number, the improper integral converges. So, for the integral to converge, the following should hold: \[ 0 < \int_{1}^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}} dx < \infty \] In this case however, \[ \int_{1}^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}} dx \], the function under the integral sign never approaches zero and therefore the integral diverges.
02

Evaluate the Diverged Integral

Since the integral has been established to diverge, evaluating it would yield a result of \(\infty\). This is because the integral \[ \int_{1}^{\infty} \frac{e^{\sqrt{x}}}{\sqrt{x}} dx \], tends towards infinity as x tends towards infinity.

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