\( \newcommand\D{\mathrm{d}} \newcommand\E{\mathrm{e}} \newcommand\I{\mathrm{i}} \newcommand\bigOh{\mathcal{O}} \newcommand{\cat}[1]{\mathbf{#1}} \newcommand\curl{\vec{\nabla}\times} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}} \)
UP | HOME

Exercises

Table of Contents

1. Advanced Integrals

1.1. Anastasios' Exercises

Taken from mathematica.gr,

Computer \(\displaystyle \int^{1/2}_{0}\frac{\ln(1-x)\ln(x)}{x(1-x)}\D x\) (Anastasios).

Let \(p\) be a positive real, \(n\in\NN_{0}\). Compute \(\displaystyle\int^{\infty}_{0}\frac{\sin^{n}(x)}{\E^{px}}\D x\).

Let \(m\in\NN_{0}\). Compute \(\displaystyle \int^{\infty}_{0}x^{m}\E^{-x}\sin(x)\,\D x\).

Let \(a > b > 0\). Compute \(\displaystyle\int^{\infty}_{0}\frac{\E^{ax}-\E^{bx}}{x(\E^{ax}+1)(\E^{bx}+1)}\D x\).

Let \(n\in\NN\), \(k=0,1,\dots,2n-2\). Define

\begin{equation} I_{k} = \int^{\infty}_{0}\frac{x^{k}}{x^{2n}+x^{n}+1}\D x. \end{equation}

Show \(I_{k}\geq I_{n-1}\) for \(k=0, 1,\dots, 2n-2\).

Compute \(\displaystyle\int^{1}_{0}\int^{1}_{0}\frac{1}{1-xy}\D x\,\D y\).

Compute \(\displaystyle\int^{1}_{0}\frac{\ln(x)\ln^{2}(1-x)}{x}\D x\).

Let \(k\in\NN\). Compute \(\displaystyle\int^{\infty}_{0}\frac{\sin(kx)\cos^{k}(x)}{x}\D x\).

How does \(\displaystyle \int^{\infty}_{0}\frac{1}{x+1}\D x\) converge or diverge?

Prove \(\displaystyle\int^{0}_{-1}\frac{\ln(1+t)}{t}\D t\) converges or diverges.

Prove the following integral converges or diverges: \(\displaystyle \int^{\E}_{1}\frac{\D x}{x\sqrt{\ln(x)}}\)

For what \(a,b\in\RR\) does the following integral converge: \(\displaystyle\int^{1}_{0}\frac{\D x}{x^{a}(-\ln(x))^{b}}\).

  1. Calculate \(\displaystyle\int^{\infty}_{0}\frac{x^{a}}{(m+x^{b})^{c}}\D x\) for \(a > -1\), \(b > 0\), \(m > 0\), \(c > (1 + a)/b\)
  2. Calculate \(\displaystyle\int^{\infty}_{0}\E^{-a^{2}x^{2}}\cos(bx)\,\D x\) where \(ab\neq0\).

Let \(b>0\), \(-\pi/2\lt\varphi\lt\pi/2\), compute \(\displaystyle\int^{\infty}_{0}\E^{-t\cos(\varphi)}t^{b-1}\sin(t\sin(\varphi))\,\D t\)

Let \(b\gt 0\) and \(-\pi/2\lt\varphi\lt\pi/2\), compute \(\displaystyle\int^{\infty}_{0}\E^{-t\cos(\varphi)}t^{b-1}\cos(t\sin(\varphi))\,\D t\)

Determine the convergence of \(\displaystyle\int^{\infty}_{0}\tan\left(\frac{x}{\sqrt{x^{2}+x^{3}}}\right)\frac{\ln(1+\sqrt{x})}{x}\D x\).

Concerning Frullani integrals. Let \(a\gt b\gt0\), \(f\colon[0,\infty)\to\RR\) be continuous such that

  1. \(\displaystyle\int^{\infty}_{1}\frac{f(t)}{t}\D t\) converges,
  2. \(f(x+T)=f(x)\) for some \(T\gt 0\) and for all \(x\geq0\)
  3. \(\displaystyle\lim_{x\to+\infty}f(x)=\ell\in\RR\).

Then calculate \(\displaystyle\int^{\infty}_{0}\frac{f(ax)-f(bx)}{x}\D x\).

Calculate the following integrals:

  1. \(\displaystyle\int^{\infty}_{0}\frac{\sin(x)}{1 + x^{2}}\D x\)
  2. \(\displaystyle\int^{\infty}_{0}\frac{\cos(x)}{1 + x^{2}}\D x\)

Determine the convergence of \(\displaystyle\int^{\infty}_{0}\frac{\D x}{1+x^{2}\sin^{2}(x)}\).

Recall \(\displaystyle\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int^{x}_{0}\E^{-t^{2}}\,\D t\). Compute \(\displaystyle\int^{\infty}_{0}\operatorname{erf}^{2}(\sqrt{x})\E^{-x}\,\D x\).

Calculate \(\displaystyle\int^{\infty}_{0}\frac{x^{2}}{2x^{4} + 5x^{2} + 2}\D x\).

Compute \(\displaystyle\int^{\infty}_{0}\left(\frac{x}{\E^{x}-\E^{-x}}-\frac{1}{2}\right)\frac{1}{x^{2}}\D x\).

Calculate \(\displaystyle\int^{\infty}_{-\infty}\E^{\I kx}\frac{1 - \E^{x}}{\E^{x} + 1}\D x\).

Let \(k\in\RR\), calculate \(\displaystyle\int^{\pi/2}_{0}\cos\bigl(k\cdot\ln(\tan(x))\bigr)\,\D x\).

Let \(n\in\NN\), calculate \(\displaystyle\int^{\infty}_{0}x^{n}\sin(\sqrt[4]{x})\E^{-\sqrt[4]{x}}\D x\).

Calculate \(\displaystyle\int^{\pi/2}_{0}x\ln(\tan(x))\,\D x\).

Calculate \(\displaystyle\int^{\pi/2}_{0}\ln^{2}(\cos(x))\,\D x\).

Let \(|a|\lt 1\). Compute \(\displaystyle\int^{\pi/2}_{0}\frac{\tan^{-1}(a\sin(x))}{\sin(x)}\D x\).

Calculate \(\displaystyle\int^{1}_{0}\frac{\sqrt{1-x^{2}}}{1 - x^{2}\sin^{2}(x)}\D x\).

Let \(m\in\RR\) and \(-1\lt a\lt 1\), compute \(\displaystyle\int^{2\pi}_{0}\frac{\E^{m\cos(\theta)}\bigl(\cos(m\sin(\theta))-a\sin(m\sin(\theta)+\theta)\bigr)}{1-2a\sin(\theta)+a^{2}}\D\theta\).

For \(0\lt a\lt\pi/2\), calculate \(\displaystyle\int^{\infty}_{-\infty}\frac{\tan^{-1}(x)}{x^{2} - 2x\sin(a)+1}\D x\).

Calculate \(\displaystyle\int^{\infty}_{0}\frac{\sin(x)}{x\E^{x}}\D x\).

Calculate \(\displaystyle\int^{\infty}_{0}\int^{\infty}_{0}\int^{\infty}_{0}\frac{\D x\,\D y\,\D z}{(x^{1} + y^{2} + z^{2} + 1)^{2}}\).

Compute \(\displaystyle\int^{\infty}_{0}\frac{x\ln(x)}{(1 + x^{2})(1 + x^{3})^{2}}\D x\).

Evaluate \(\displaystyle\int^{\infty}_{0}\frac{\sqrt{x}\ln(x)}{1 + x^{2}}\D x\).

Let \(a_{1},\dots, a_{n}, b_{1},\dots, b_{m}, b\in\RR\) such that

\begin{equation} b\gt\sum^{n}_{i=1}|a_{i}| + \sum^{m}_{j=1}|b_{j}| \end{equation}

Calculate \(\displaystyle\int^{\infty}_{0}\left(\prod^{n}_{i=1}\frac{\sin(a_{i}x)}{x}\right)\left(\prod^{m}_{j=1}\cos(b_{j}x)\right)\frac{\sin(bx)}{x}\D x\).

Compute \(\displaystyle\int^{\infty}_{0}\frac{\cos(x^{2})-\cos(x)}{x}\D x\).

Prove or find a counter-example

\begin{equation} \int^{\infty}_{0}\frac{(1 + z)^{-n}}{\log^{2}(z) + \pi^{2}}\D z = (-1)^{n-1}\int^{1}_{0}\binom{t}{n}\, \D t \end{equation}

where \(n\in\NN\), and \[ \binom{t}{n} = \frac{t(t-1)(\dots)(t - n + 1)}{n!} \]

Compute \(\displaystyle\int^{\infty}_{-\infty}\frac{x\sin(x)}{x^{2}+4x+20}\D x\).

Let \(\lambda\in\RR\), compute \(\displaystyle\int^{\infty}_{0}\frac{4\cos(\lambda x)}{\E^{x} + \E^{-x}}\D x\).

Calculate \(\displaystyle\int^{\infty}_{-\infty}\frac{\E^{-\I x}}{x^{2}-2x+4}\D x\).

Compute \(\displaystyle\int^{\infty}_{0}\frac{x}{(x^{2}+1)(x^{4}+x^{2}+1)}\D x\).

Calculate \(\displaystyle\int^{\infty}_{0}\frac{1}{(x^{4} + (1 + \sqrt{2})x^{2} + 1)(x^{100}-x^{98}+\dots+1)}\D x\).

Let \(n\in\NN\) and \(p\gt0\), calculate \(\displaystyle\int^{\infty}_{0}x^{2n}\E^{-px^{2}}\,\D x\).

Calculate \(\displaystyle\int^{\infty}_{-\infty}\left(1 + \frac{x^{2}}{n-1}\right)^{-n/2}\D x\).

Calculate \(\displaystyle\int^{\pi/2}_{0}x^{2}\ln^{2}(2\cos(x))\,\D x\).

Calculate \(\displaystyle\int^{\infty}_{0}\frac{x^{2}}{x^{4} + 30x^{2} + 1}\D x\).

Compute \(\displaystyle\int^{\infty}_{0}\frac{1}{(x^{4} + 2ax^{2} + 1)^{2}}\D x\).

Evaluate \(\displaystyle\int^{\infty}_{0}\frac{x^{29}}{(5x^{2}+49)^{17}}\D x\).

Evaluate \(\displaystyle\int^{\infty}_{0}\frac{3x^{3}}{(x^{4} + 4x^{2}+1)^{5}}\D x\).

\(\displaystyle\int^{\pi/2}_{0}\ln(\cos(x))\ln(\sin(x))\sin(2x)\,\D x\).

\(\displaystyle\int^{1}_{0}\frac{\ln(x)}{x^{2}-x-1}\D x\).

Prove, if \(a\in\CC\) is such that \(0\lt|\operatorname{Im}(a)|<2\pi\), then \(\displaystyle\int^{\infty}_{0}\frac{\sin(ax)}{\exp(2\pi x)-1}\D x=\frac{-1}{2a^{2}}+\frac{1}{4}\frac{\E^{a}+1}{\E^{a}-1}\).

Let \(\lambda\in\RR\), calculate \(\displaystyle\int^{\infty}_{-\infty}x\E^{-\lambda x^{2}}\,\D x\).

Prove \(2\sqrt{2}\log(1 + \sqrt{2}) = \displaystyle\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\operatorname{sgn}(x)\operatorname{sgn}(y)\E^{-(x^{2}+y^{2})/2}\sin(xy)\,\D x\,\D y\) where \[ \operatorname{sgn}(x) = \begin{cases} 1, & x\gt0\\ 0, & x=0\\ -1, & x\lt0\end{cases} \]

Compute \(\displaystyle\int^{\infty}_{0}\frac{\cos(t^{2}) - \sin(t^{2})}{1 + t^{4}}\D t\).

Let \(a\gt 0\), compute \(\displaystyle\int^{\infty}_{-\infty}\frac{\cos(x)-1}{x^{2}(x^{2}+a)}\D x\).

Prove

\begin{equation} \int^{\infty}_{0}\prod^{\infty}_{n=1}\frac{1 + \left(\frac{x}{b+n}\right)^{2}}{1 + \left(\frac{x}{a+n-1}\right)^{2}}\D x = \frac{\sqrt{\pi}}{2}\frac{\Gamma(a + 1/2)}{\Gamma(a)}\frac{\Gamma(b+1)}{\Gamma(b + 1/2)}\frac{\Gamma(b-a+1/2)}{\Gamma(b-a+1)} \end{equation}

Calculate \(\displaystyle\int^{\infty}_{0}\frac{\E^{-ax^{2}}-1}{x^{2}}\D x\) and determine where it converges.

Calculate \(\displaystyle\int^{\infty}_{-\infty}\frac{(3x+1)\sin(ax)}{9x^{2}+6x+10}\D x\) and determine where it converges.

Calculate \(\displaystyle\int^{\infty}_{0}\frac{\ln(2x)}{x^{2}+9}\D x\).

Let \(n\in\NN\) be greater than 1, calculate \(\displaystyle\int^{\infty}_{0}\frac{1}{x^{n}+1}\D x\).

Last Updated 2021-06-01 Tue 10:00.