Verla
MathCalculus

Evaluate the Integral or State That It Diverges: Six Improper Integrals

Rewriting each improper integral as a limit gives ∫₁^∞ dx/x^e = 1/(e−1), ∫₀¹ dx/√(1−x) = 2 and ∫₀^∞ x e^{−x} dx = 1, while ∫₀¹ dx/x^π, ∫₀¹ dx/(1−x) and ∫₀¹ (ln x)/x dx all diverge.

Question

Evaluate the integrals. If the integral diverges, answer "It diverges."

(a) ∫1∞dxxe\displaystyle \int_{1}^{\infty} \frac{dx}{x^{e}}

(b) ∫01dxxπ\displaystyle \int_{0}^{1} \frac{dx}{x^{\pi}}

(c) ∫01dx1−x\displaystyle \int_{0}^{1} \frac{dx}{\sqrt{1-x}}

(d) ∫01dx1−x\displaystyle \int_{0}^{1} \frac{dx}{1-x}

(e) ∫0∞xe−x dx\displaystyle \int_{0}^{\infty} x e^{-x}\,dx

(f) ∫01ln⁡xx dx\displaystyle \int_{0}^{1} \frac{\ln x}{x}\,dx

This question has been answered.

Explore the full answer and supporting references to better understand this question.

Verla BasicBest value

$19.99/month

  • Everything in Study Pass
  • Assignments with writing output
  • AI detection and humanizer
  • Basic quotas across every tool
Study Pass

$0.99/month

  • Entire question library
  • Knowledge guides and sources
  • Detailed answers and explanations
  • No AI tools or quotas

Evidence boundary

This page is a normalized presentation of the OpenStax Calculus Volume 2 section 3.7 exercise set that reads “Evaluate the integrals. If the integral diverges, answer ‘It diverges.’” Six of the eleven items in that set are used, and each integrand, pair of limits and singular point is reproduced as published; nothing is reconstructed from the truncated keyword. The page reports the convergence or divergence of these six integrals only. It does not prove the comparison theorem, and it makes no claim about series built from the same integrands.