https://file.aiquickdraw.com/mathbot/file/1730312296746bdsy2ysb.jpg

To evaluate the improper integral 

\[
\int_{-\infty}^{-1} x^{-\frac{8}{3}} \, dx,
\]

let's go through the steps to determine if this integral converges and find its value if possible.

### Step 1: Set up the Integral
The given function \( x^{-\frac{8}{3}} \) has a negative exponent, making it undefined at \( x = 0 \), and potentially divergent as \( x \to -\infty \). Since the interval of integration is \( (-\infty, -1] \), we are dealing with an improper integral due to the unbounded lower limit. 

Rewrite the integral with a limit to handle the improper bound:

\[
\int_{-\infty}^{-1} x^{-\frac{8}{3}} \, dx = \lim_{a \to -\infty} \int_{a}^{-1} x^{-\frac{8}{3}} \, dx.
\]

### Step 2: Integrate \( x^{-\frac{8}{3}} \)
To integrate \( x^{-\frac{8}{3}} \), recall the power rule for integration:

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad \text{for } n \neq -1.
\]

In this case, \( n = -\frac{8}{3} \), so:

\[
n + 1 = -\frac{8}{3} + 1 = -\frac{5}{3}.
\]

Thus,

\[
\int x^{-\frac{8}{3}} \, dx = \frac{x^{-\frac{5}{3}}}{-\frac{5}{3}} = -\frac{3}{5} x^{-\frac{5}{3}}.
\]

### Step 3: Apply the Limits
Now, substitute back into the integral:

\[
\lim_{a \to -\infty} \int_{a}^{-1} x^{-\frac{8}{3}} \, dx = \lim_{a \to -\infty} \left[ -\frac{3}{5} x^{-\frac{5}{3}} \right]_{a}^{-1}.
\]

Evaluating this at the bounds:

\[
= \lim_{a \to -\infty} \left( -\frac{3}{5} \cdot (-1)^{-\frac{5}{

Evaluate the improper integral \( \int_{-\infty}^{-1} x^{-\frac{8}{3}} \, dx \) with detailed steps.

Learn how to evaluate the improper integral \( \int_{-\infty}^{-1} x^{-8/3} \, dx \) with a step-by-step solution. This problem involves calculus concepts like improper integrals and power rule for integration, suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation