Problem Set - Improper integrals

## Problem Set - Improper integrals

Last updates: 7 December 2009

## Improper integrals

 Show that if $z=\sqrt{\frac{4+x}{1-x}}$ then ${\int }_{-1}^{1}\sqrt{\frac{1+x}{1-x}}dx={\int }_{0}^{\infty }\frac{4{z}^{2}}{{\left({z}^{2}+1\right)}^{2}}dz$. 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 ${\int }_{-1}^{1}\sqrt{\frac{1+x}{1-x}}dx=\pi$. Show that ${\int }_{0}^{1}\frac{1}{x}dx$ diverges. Evaluate ${\int }_{0}^{3}\frac{dx}{{\left(x-1\right)}^{2/3}}$. Determine whether ${\int }_{1}^{\infty }\frac{1}{x}dx$ converges or diverges. Determine whether ${\int }_{1}^{\infty }\frac{1}{{x}^{2}}dx$ converges or diverges. Determine whether ${\int }_{1}^{\infty }{e}^{-{x}^{2}}dx$ converges or diverges. Show that ${\int }_{1}^{\infty }\frac{1}{{x}^{p}}dx$ converges if $p\in ℝ$ and $p>1$. Show that ${\int }_{1}^{\infty }\frac{1}{{x}^{p}}dx$ diverges if $p\in ℝ$ and $p\le 1$. Evaluate ${\int }_{0}^{\infty }\frac{1}{{x}^{2}+1}dx$. Evaluate ${\int }_{0}^{1}\frac{1}{\sqrt{x}}dx$. Evaluate ${\int }_{-1}^{1}\frac{1}{{x}^{2/3}}dx$. Evaluate ${\int }_{1}^{\infty }\frac{1}{{x}^{1.001}}dx$. Evaluate ${\int }_{0}^{4}\frac{1}{\sqrt{4-x}}dx$. Evaluate ${\int }_{0}^{1}\frac{1}{\sqrt{1-{x}^{2}}}dx$. Evaluate ${\int }_{0}^{\infty }{e}^{-x}\mathrm{cos}x\text{\hspace{0.17em}}dx$. Evaluate ${\int }_{0}^{1}\frac{1}{{x}^{0.999}}dx$. Determine whether ${\int }_{1}^{\infty }\frac{1}{\sqrt{x}}dx$ converges or diverges. Determine whether ${\int }_{1}^{\infty }\frac{1}{{x}^{3}}dx$ converges or diverges. Determine whether ${\int }_{1}^{\infty }\frac{1}{{x}^{3}+1}dx$ converges or diverges. Determine whether ${\int }_{0}^{\infty }\frac{1}{{x}^{3}}dx$ converges or diverges. Determine whether ${\int }_{0}^{\infty }\frac{1}{{x}^{3}+1}dx$ converges or diverges. Determine whether ${\int }_{0}^{\infty }\frac{1}{1+{e}^{x}}dx$ converges or diverges. Determine whether ${\int }_{0}^{\pi /2}\mathrm{tan}x\text{\hspace{0.17em}}dx$ converges or diverges. Determine whether ${\int }_{-1}^{1}\frac{1}{{x}^{2}}dx$ converges or diverges. Determine whether ${\int }_{-1}^{1}\frac{1}{{x}^{2/5}}dx$ converges or diverges. Determine whether ${\int }_{0}^{\infty }\frac{1}{\sqrt{x}}dx$ converges or diverges. Determine whether ${\int }_{0}^{\infty }\frac{1}{\sqrt{x+{x}^{4}}}dx$ converges or diverges. Classify the following improper integrals and evaluate them if they converge: (i)   ${\int }_{1}^{5}\frac{4x}{\sqrt{{x}^{2}-1}}$. (ii)   ${\int }_{1}^{\infty }\frac{1}{1+{x}^{2}}$. (iii)   Does the following integral diverge or converge? Explain why, but do not evaluate the integral. ${\int }_{1}^{\infty }\frac{{x}^{2}}{\left(x-2\right){\left({x}^{11}+2\right)}^{1/4}}$.