Questions: Determine whether the improper integral ∫ from 1 to ∞ of x^(-3/2) dx converges or diverges. If it converges, find its value. If it diverges, enter DNE. Converges or Diverges: □ ∫ from 1 to ∞ of x^(-3/2) dx = □

Determine whether the improper integral ∫ from 1 to ∞ of x^(-3/2) dx converges or diverges. If it converges, find its value. If it diverges, enter DNE.

Converges or Diverges: □

∫ from 1 to ∞ of x^(-3/2) dx =

□

Transcript text: Determine whether the improper integral $\int_{1}^{\infty} x^{-\frac{3}{2}} d x$ converges or diverges. If it converges, find its value. If it diverges, enter DNE. Converges or Diverges: $\square$ \[ \int_{1}^{\infty} x^{-\frac{3}{2}} d x= \] $\square$ Video Example: Solving A Similar Problem

Solution

Solution Steps

To determine whether the improper integral converges or diverges, we can evaluate the integral from 1 to infinity of $ x^{-\frac{3}{2}} $. If the result is a finite number, the integral converges; otherwise, it diverges. We can use the power rule for integration to find the antiderivative and then evaluate the limit as the upper bound approaches infinity.

Step 1: Determine the Convergence of the Integral

To determine whether the integral $\int_{1}^{\infty} x^{-\frac{3}{2}} \, dx$ converges, we evaluate the integral using the power rule for integration. The antiderivative of $x^{-\frac{3}{2}}$ is:

\[ \int x^{-\frac{3}{2}} \, dx = \frac{x^{-\frac{1}{2}}}{-\frac{1}{2}} = -2x^{-\frac{1}{2}} \]

Step 2: Evaluate the Improper Integral

Evaluate the definite integral from 1 to $\infty$:

\[ \lim_{b \to \infty} \left[-2x^{-\frac{1}{2}}\right]_{1}^{b} = \lim_{b \to \infty} \left(-2b^{-\frac{1}{2}} + 2 \cdot 1^{-\frac{1}{2}}\right) \]

Simplifying, we have:

\[ \lim_{b \to \infty} \left(-2b^{-\frac{1}{2}} + 2\right) = 0 + 2 = 2 \]

Since the result is a finite number, the integral converges.