Improper Integrals


A. Explain why each of the following integrals is improper.   solution

(a) \(\displaystyle \int_0^\infty \frac{1}{1+x^2} \ \mathrm{d}x\)   (b) \(\displaystyle \int_5^6 \frac{1}{x-5} \ \mathrm{d}x\)   (c) \(\displaystyle \int_0^{\pi/4} \cot (x) \ \mathrm{d}x\)   (d) \(\displaystyle \int_{-\infty}^{\infty} x^2\, e^{-x^2} \ \mathrm{d}x\)

B. Using the \(p\)-Power Test, determine whether each integral converges or diverges. Do not evaluate the integral.   solution

(a) \(\displaystyle \int_1^\infty \frac{1}{x^2} \ \mathrm{d}x\)   (b) \(\displaystyle \int_1^\infty \frac{1}{x^{2/3}} \ \mathrm{d}x\)   (c) \(\displaystyle \int_1^\infty \frac{2}{x^7} \ \mathrm{d}x\)   (d) \(\displaystyle \int_1^\infty \frac{1}{x^{-15}} \ \mathrm{d}x\)

C. Determine whether each of the following improper integrals converges or diverges. If it converges, determine its value.

  1. \(\displaystyle \int_3^\infty e^{-3p} \ \mathrm{d}p\)   solution

  2. \(\displaystyle \int_0^\infty 6e^{-2x} \ \mathrm{d}x\)   solution

  3. \(\displaystyle \int_0^\infty \frac{1}{\sqrt[3]{x+2}} \ \mathrm{d}x\)   solution

  4. \(\displaystyle \int_0^\infty \frac{1}{(x+2)^3} \ \mathrm{d}x\)   solution

  5. \(\displaystyle \int_0^\infty \frac{1}{(2x+5)^3} \ \mathrm{d}x\)   solution

  6. \(\displaystyle \int_0^\infty \frac{1}{1+x^2} \ \mathrm{d}x\)   solution

  7. \(\displaystyle \int_{-\infty}^0 \frac{1}{6-9x} \ \mathrm{d}x\)   solution

  8. \(\displaystyle \int_0^\infty \frac{x^2}{5+x^3} \ \mathrm{d}x\)   solution

  9. \(\displaystyle \int_3^\infty \frac{1}{(x-2)^2} \ \mathrm{d}x\)   solution

  10. \(\displaystyle \int_4^8 \frac{1}{(x-4)^3} \ \mathrm{d}x\)   solution

  11. \(\displaystyle \int_1^{4} \frac{1}{\sqrt{x-1}} \ \mathrm{d}x\)   solution

  12. \(\displaystyle \int_1^5 \frac{1}{(x-3)^{2/3}} \ \mathrm{d}x\)   solution

  13. \(\displaystyle \int_1^6 \frac{1}{x-3} \ \mathrm{d}x\)   solution

  14. \(\displaystyle \int_4^{\infty} \frac{1}{x^2+x} \ \mathrm{d}x\)   solution