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Comparison test for improper integrals practice problems

Decide whether an improper integral converges by comparing it with a simpler one. 13 problems with worked solutions; in 13 of them every equation is proved by a computer algebra system.

4 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 4.1094

medium
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{\sin^{2}{\left(x \right)}}{x^{2}} \, dx \) converges or diverges.
  1. sin²x ≤ 1, so the integrand is at most 1/x². Both are positive.
    Reviewed
  2. \[ - \frac{\sin^{2}{\left(x \right)}}{x^{2}} + \frac{1}{x^{2}} = \frac{\cos^{2}{\left(x \right)}}{x^{2}} \]
    The difference, which is ≥ 0 for x ≥ 1.✓ Proved
  3. \[ \int\limits_{1}^{\infty} \frac{1}{x^{2}}\, dx = 1 \]
    The comparison integral converges.✓ Proved
  4. A smaller positive integrand than a convergent one converges; a larger one than a divergent one diverges. So it converges.
    Reviewed
Answer \( \text{converges} \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1095

medium
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{x + 3}{x^{2}} \, dx \) converges or diverges.
  1. (x + 3)/x² = 1/x + 3/x² ≥ 1/x.
    Reviewed
  2. \[ - \frac{1}{x} + \frac{x + 3}{x^{2}} = \frac{3}{x^{2}} \]
    The difference, which is ≥ 0 for x ≥ 1.✓ Proved
  3. \[ \int\limits_{1}^{T} \frac{1}{x}\, dx = \ln{\left(T \right)} \]
    Up to T the comparison integral is this, which grows without bound as T → ∞: it diverges.✓ Proved
  4. A smaller positive integrand than a convergent one converges; a larger one than a divergent one diverges. So it diverges.
    Reviewed
Answer \( \text{diverges} \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1099

medium
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{1}{x^{2} + 5 x} \, dx \) converges or diverges.
  1. x² + 5x ≥ x² for x ≥ 1, so the integrand is at most 1/x². Both are positive.
    Reviewed
  2. \[ - \frac{1}{x^{2} + 5 x} + \frac{1}{x^{2}} = \frac{5}{x^{2} \left(x + 5\right)} \]
    The difference, which is ≥ 0 for x ≥ 1.✓ Proved
  3. \[ \int\limits_{1}^{\infty} \frac{1}{x^{2}}\, dx = 1 \]
    The comparison integral converges.✓ Proved
  4. A smaller positive integrand than a convergent one converges; a larger one than a divergent one diverges. So it converges.
    Reviewed
Answer \( \text{converges} \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 4.1100

medium
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{1}{x^{2} + x} \, dx \) converges or diverges.
  1. x² + 1x ≥ x² for x ≥ 1, so the integrand is at most 1/x². Both are positive.
    Reviewed
  2. \[ - \frac{1}{x^{2} + x} + \frac{1}{x^{2}} = \frac{1}{x^{2} \left(x + 1\right)} \]
    The difference, which is ≥ 0 for x ≥ 1.✓ Proved
  3. \[ \int\limits_{1}^{\infty} \frac{1}{x^{2}}\, dx = 1 \]
    The comparison integral converges.✓ Proved
  4. A smaller positive integrand than a convergent one converges; a larger one than a divergent one diverges. So it converges.
    Reviewed
Answer \( \text{converges} \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More comparison test for improper integrals practice

Use a comparison to decide whether \( \displaystyle \int_1^\infty e^{- x^{2}} \, dx \) converges or diverges.
Problem 4.1092medium✓ Every equation proved
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{1}{\sqrt{x} - \frac{1}{2}} \, dx \) converges or diverges.
Problem 4.1093medium✓ Every equation proved
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{x + 5}{x^{2}} \, dx \) converges or diverges.
Problem 4.1096medium✓ Nihil obstat
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{1}{x^{2} + 2 x} \, dx \) converges or diverges.
Problem 4.1097medium✓ Every equation proved
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{x + 4}{x^{2}} \, dx \) converges or diverges.
Problem 4.1098medium✓ Nihil obstat
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{x + 2}{x^{2}} \, dx \) converges or diverges.
Problem 4.1101medium✓ Nihil obstat
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{1}{x^{2} + 4 x} \, dx \) converges or diverges.
Problem 4.1102medium✓ Every equation proved
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{1}{x^{2} + 3 x} \, dx \) converges or diverges.
Problem 4.1103medium✓ Nihil obstat
Use a comparison to decide whether \( \displaystyle \int_1^\infty \frac{x + 1}{x^{2}} \, dx \) converges or diverges.
Problem 4.1104medium✓ Nihil obstat