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.
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\(\displaystyle \int_3^\infty e^{-3p} \ \mathrm{d}p\) solution
-
\(\displaystyle \int_0^\infty \frac{1}{\sqrt[3]{x+2}} \ \mathrm{d}x\)
solution
-
\(\displaystyle \int_0^\infty \frac{1}{(x+2)^3} \ \mathrm{d}x \)
solution
-
\(\displaystyle \int_0^\infty \frac{1}{1+x^2} \ \mathrm{d}x\)
solution
-
\(\displaystyle \int_{-\infty}^0 \frac{1}{6-9x} \ \mathrm{d}x\) solution
-
\(\displaystyle \int_4^8 \frac{1}{(x-4)^3} \ \mathrm{d}x\)
solution
-
\(\displaystyle \int_3^\infty \frac{1}{(x-2)^2} \
\mathrm{d}x\) solution
-
\(\displaystyle \int_1^5 \frac{1}{(x-3)^{2/3}} \ \mathrm{d}x \)
solution
-
\(\displaystyle \int_1^{4} \frac{1}{\sqrt{x-1}}\, \mathrm{d}x \)
solution
-
\(\displaystyle \int_1^6 \frac{1}{x-3} \ \mathrm{d}x \)
solution
-
\(\displaystyle \int_0^\infty \frac{x^2}{5+x^3} \ \mathrm{d}x\) solution
