Problem Set - Improper integrals

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Improper integrals

	Show that if $z=\sqrt{{\displaystyle \frac{4+x}{1-x}}}$ then ${\displaystyle {\int }_{-1}^1\sqrt{\frac{1+x}{1-x}}}dx={\displaystyle {\int }_0^{\infty }\frac{4z^2}{{(z^2+1)}^2}}dz$. The improper integral on the left is an improper integral of the first kind and the improper integral on the right is an improper integral of the second kind.
	Show that ${\displaystyle {\int }_{-1}^1\sqrt{\frac{1+x}{1-x}}}dx=\pi$.
	Show that ${\displaystyle {\int }_0^1\frac{1}{x}dx}$ diverges.
	Evaluate ${\displaystyle {\int }_0^3\frac{dx}{{(x-1)}^{\mathrm{2/3}}}}$.
	Determine whether ${\displaystyle {\int }_1^{\infty }\frac{1}{x}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_1^{\infty }\frac{1}{x^2}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_1^{\infty }{e}^{-x^2}dx}$ converges or diverges.
	Show that ${\displaystyle {\int }_1^{\infty }\frac{1}{x^p}dx}$ converges if $p\in ℝ$ and $p>1$.
	Show that ${\displaystyle {\int }_1^{\infty }\frac{1}{x^p}dx}$ diverges if $p\in ℝ$ and $p\le 1$.
	Evaluate ${\displaystyle {\int }_0^{\infty }\frac{1}{x^2+1}dx}$.
	Evaluate ${\displaystyle {\int }_0^1\frac{1}{\sqrt{x}}dx}$.
	Evaluate ${\displaystyle {\int }_{-1}^1\frac{1}{x^{\mathrm{2/3}}}dx}$.
	Evaluate ${\displaystyle {\int }_1^{\infty }\frac{1}{x^{1.001}}dx}$.
	Evaluate ${\displaystyle {\int }_0^4\frac{1}{\sqrt{4-x}}dx}$.
	Evaluate ${\displaystyle {\int }_0^1\frac{1}{\sqrt{1-x^2}}dx}$.
	Evaluate ${\displaystyle {\int }_0^{\infty }{e}^{-x}\cos xdx}$.
	Evaluate ${\displaystyle {\int }_0^1\frac{1}{x^{0.999}}dx}$.
	Determine whether ${\displaystyle {\int }_1^{\infty }\frac{1}{\sqrt{x}}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_1^{\infty }\frac{1}{x^3}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_1^{\infty }\frac{1}{x^3+1}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_0^{\infty }\frac{1}{x^3}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_0^{\infty }\frac{1}{x^3+1}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_0^{\infty }\frac{1}{1+e^x}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_0^{\pi /2}\tan xdx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_{-1}^1\frac{1}{x^2}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_{-1}^1\frac{1}{x^{\mathrm{2/5}}}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_0^{\infty }\frac{1}{\sqrt{x}}dx}$ converges or diverges.
	Determine whether ${\displaystyle {\int }_0^{\infty }\frac{1}{\sqrt{x+x^4}}dx}$ converges or diverges.
	Classify the following improper integrals and evaluate them if they converge: (i)   ${\displaystyle {\int }_1^5\frac{4x}{\sqrt{x^2-1}}}$. (ii)   ${\displaystyle {\int }_1^{\infty }\frac{1}{1+x^2}}$. (iii)   Does the following integral diverge or converge? Explain why, but do not evaluate the integral. ${\displaystyle {\int }_1^{\infty }\frac{x^2}{(x-2){(x^{11}+2)}^{\mathrm{1/4}}}}$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)