Question

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

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

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.

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.