Improper Integrals

A. Explain why each of the following integrals is improper. Watch Explanation ↗

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

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

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

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

1. \(\displaystyle \int_3^\infty e^{-3p} \ dp\) Solution ↗
2. \(\displaystyle \int_0^\infty 6e^{-2x} \ dx\) Solution ↗
3. \(\displaystyle \int_3^\infty 12e^{-2x} \ dx\) Solution (PDF) ↗
4. \(\displaystyle \int_0^\infty \frac{1}{\sqrt[3]{x+2}} \ dx\) Solution ↗
5. \(\displaystyle \int_0^\infty \frac{1}{(x+2)^3} \ dx\) Solution ↗
6. \(\displaystyle \int_0^\infty \frac{1}{(2x+5)^3} \ dx\) Solution ↗
7. \(\displaystyle \int_0^\infty \frac{1}{1+x^2} \ dx\) Solution ↗
8. \(\displaystyle \int_{-\infty}^0 \frac{1}{6-9x} \ dx\) Solution ↗
9. \(\displaystyle \int_0^\infty \frac{x^2}{5+x^3} \ dx\) Solution (PDF) ↗
10. \(\displaystyle \int_0^\infty \frac{x^2}{\sqrt{6+x^3}} \ dx\) Solution ↗
11. \(\displaystyle \int_3^\infty \frac{1}{(x-2)^2} \ dx\) Solution ↗
12. \(\displaystyle \int_2^\infty \frac{e^{-1/x}}{x^2} \ dx\) Solution ↗
13. \(\displaystyle \int_4^8 \frac{1}{(x-4)^3} \ dx\) Solution ↗
14. \(\displaystyle \int_1^{4} \frac{1}{\sqrt{x-1}} \ dx\) Solution ↗
15. \(\displaystyle \int_1^5 \frac{1}{(x-3)^{2/3}} \ dx\) Solution ↗
16. \(\displaystyle \int_1^6 \frac{1}{x-3} \ dx\) Solution (Image) ↗
17. \(\displaystyle \int_4^{\infty} \frac{1}{x^2+x} \ dx\) Solution ↗