https://file.aiquickdraw.com/mathbot/file/17266693671015l9e7qnl.jpg

The image contains three improper integrals, and the task is to determine whether each of them converges or diverges. Let's analyze them one by one.

### 1. \(\int_0^{+\infty} \frac{x}{\sqrt{x^2 + 1}} \, dx\)
This integral involves the function \(\frac{x}{\sqrt{x^2 + 1}}\), which behaves similarly to \(\frac{x}{x} = 1\) for large values of \(x\). At \(x \to \infty\), the function approaches 1, so the integral behaves like \(\int_1^{+\infty} 1 \, dx\), which diverges. Therefore, **this integral diverges**.

### 2. \(\int_0^{+\infty} \frac{x}{x^2 + 1} \, dx\)
This integral can be solved using substitution:
Let \( u = x^2 + 1 \), hence \( du = 2x \, dx \).
This substitution leads to a known form of the arctangent function, and it converges. Therefore, **this integral converges**.

### 3. \(\int_0^{+\infty} \frac{x^3}{\sqrt{x^4 + 1}} \, dx\)
For large values of \(x\), the function \(\frac{x^3}{\sqrt{x^4 + 1}}\) behaves like \(\frac{x^3}{x^2} = x\). The integral of \(x\) diverges as \(x \to \infty\), so **this integral diverges**.

### Summary:
1. \(\int_0^{+\infty} \frac{x}{\sqrt{x^2 + 1}} \, dx\) → **Diverges**.
2. \(\int_0^{+\infty} \frac{x}{x^2 + 1} \, dx\) → **Converges**.
3. \(\int_0^{+\infty} \frac{x^3}{\sqrt{x^4 + 1}} \, dx\) → **Diverges**.

Would you like more details on how the convergence or divergence is determined?

Here are 5 related questions for further exploration:
1. How can you use comparison tests to analyze the convergence of improper integrals?
2. What are the conditions for the integral of a rational function to converge?
3. How does substitution help in solving integrals involving square roots?
4. Can you apply the limit comparison test to integrals involving powers of \(x\)?
5. How does the behavior of a function at infinity affect the convergence of improper integrals?

**Tip**: When working with improper integrals, always consider the behavior of the integrand as \(x \to \infty\) and \(x \to 0\), as these limits often determine convergence.

Установите сходимость или расходимость интегралов: 
1. ∫(x/(sqrt(x^2+1))) dx from 0 to +∞ 
2. ∫(x/(x^2+1)) dx from 0 to +∞ 
3. ∫(x^3/(sqrt(x^4+1))) dx from 0 to +∞

This problem analyzes the convergence or divergence of three improper integrals involving rational and radical expressions. Using methods like substitution and comparison tests, we determine the limits of the integrals for large values of x.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation