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11 votes
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Evaluate $\int_{0}^{1} \frac{\ln x}{(1+8x^2)\sqrt{1-x^2}} dx$

$$I=\int_0^1\frac{log(x)}{(1 + 8 x^2) \sqrt{1 - x^2}}\,dx$$ Put $x=\cos\theta$ $$I=\int_0^\frac\pi2 \frac{ln(\cos\theta)}{1+8\cos^2\theta}\,d\theta$$ Put $t=\tan\theta$ $$I=\int_0^{\infty}\frac{ln\...
whatamidoing's user avatar
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6 votes
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How to evaluate $\int_0^{\infty } e^{-x^2+2 x} \text{erf}(x+1) \, dx$

The problem can be reduced to $$I= \frac{1}{\sqrt{\pi}} \left(\text{e}\pi - f(1,1) - \frac{1}{\text{e}} f(2,\infty) \right), $$ where $f(a,b) = \int_0^b \text{e}^{-ax^2} \frac{1}{1+x^2}\text{d}x$. ...
Noctis's user avatar
  • 426
4 votes

Computing the integral $ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi. $

I suppose that a trick would be to use $$\log \left(1-2 x \cos (\phi)+x^2\right)=\sum_{n=1}^\infty \frac{ \cos (n \phi )}{n}\, x^n$$ Using Euler representation of the cosine function $$J_n=\Re\Big(\...
Claude Leibovici's user avatar
4 votes

Integrate $\frac1{\sqrt{u^2+v^2} \left(1+u^2+v^2\right)}$ over $[-1,1]^2$?

$$\int\frac{du}{\sqrt{u^2+v^2}\left(1+u^2+v^2\right)}=\frac{1}{\sqrt{v^2+1}}\tanh ^{-1}\left(\frac{u^2+v^2+1-u \sqrt{u^2+v^2}}{\sqrt{v^2+1}}\right)$$ Use the bounds and the logarithmic form of the ...
Claude Leibovici's user avatar
3 votes

How to evaluate $\int_0^{\infty } e^{-x^2+2 x} \text{erf}(x+1) \, dx$

I'm just gonna solve the last two integrals that @Noctis got to. For the first integral we have $$\begin{aligned} \int_0^1 e^{-x^2}\dfrac{\mathrm dx}{1+x^2}&=\sum_{n\geq0}\dfrac{(-1)^n}{n!}\int_0^...
Conreu's user avatar
  • 2,523
3 votes

Is it possible that : $\int_{a}^{b}e^{\ln(x)}dx=\int_{a}^{b}xdx$?

Consider these simple facts from basic analysis: $\ln(x)$ is an application from $(0, +\infty)$ to $\mathbb{R}$. The exponential $e^x$ is an application from $\mathbb{R}$ to $(0, +\infty)$. It's then ...
Enrico M.'s user avatar
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3 votes

Integral Representation of the Dottie Number

Here I present a result with the Burniston-Siewert method. First, the equation \begin{equation} \cos\left(w\right)-w=0 \end{equation} we transform it by changing the variable $w\rightarrow\textrm{...
Francisco Alvarado's user avatar
2 votes

Evaluate $\int_{0}^{1} \frac{\ln x}{(1+8x^2)\sqrt{1-x^2}} dx$

Starting with OP’s substitution, $x=\sin \theta$, then $$ I=\int_0^{\frac{\pi}{2}} \frac{\ln (\sin \theta)}{1+8 \sin ^2 \theta} d \theta $$ Let $t=\cot \theta$, we have $$ \begin{aligned} I & =\...
Lai's user avatar
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2 votes

Evaluate $\int_{0}^{1} \frac{\ln x}{(1+8x^2)\sqrt{1-x^2}} dx$

Let $t=\frac{\sqrt {1-x^2}}{x} $, then $x^2=\frac{1}{1+t^2}$ and $$ \begin{aligned}\int_0^1 \frac{\ln x}{\left(1+8 x^2\right) \sqrt{1-x^2}} d x = & \int_{\infty}^0 \frac{\ln \left(\frac{1}{\sqrt{...
Lai's user avatar
  • 22.3k
2 votes

How to evaluate $\int_0^{\infty } e^{-x^2+2 x} \text{erf}(x+1) \, dx$

Thank you all for sharing the nice analyses. I was able to evaluate the first integral in my original post. First integral: $$e \int_{-\sqrt{2}}^0 \frac{\exp \left(-\frac{y^2}{2}\right) \text{erf}\...
mattTheMathLearner's user avatar
2 votes

Computing the integral $ \int_0^{\infty} e^{-\phi^2+\phi}\cdot \phi^{2} \ln(1-2x\cos\phi+x^2)\, d\phi. $

Restatement of the problem. Firstly, the logarithm can be split thanks to the given factorization, namely $(1 - 2x\cos\phi + x^2) = (1-xe^{i\phi})(1-xe^{-i\phi})$. Secondly, the factor $\phi^2$ can be ...
Abezhiko's user avatar
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2 votes
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Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$

Let’s consider the hyperbolic function substitution $x=\cosh(i\theta)$. Then our indefinite integral is transformed into $$ \begin{aligned} I & = \int \ln \left(\frac{x+1}{x-1}\right)\left(x-\...
Lai's user avatar
  • 22.3k
1 vote
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Why the "$+1$" in $\sum_{x=\lceil u\rceil}^ne^{-f(x)}\leq\int_u^\infty e^{-f(x)} \; dx + 1$?

If $f$ is monotonically increasing and the sum is convergent, then \begin{align*} \sum\limits_{x = \left\lceil u \right\rceil }^n {{\rm e}^{ - f(x)} } & = {\rm e}^{ - f(\left\lceil u \right\rceil )...
Gary's user avatar
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1 vote

Integration of function $ [ \int_{0}^{\pi} |\sin x - \cos x| \, dx ] $

Step 1) Rewrite absolute value: \begin{equation} |\sin(x) - \cos(x)| = \sqrt{\left(\sin(x) - \cos(x)\right)^{2}} = \sqrt{1-\sin(2x)} \end{equation} Step 2) Get indefinite integral The integral of $\...
Dennis Marx's user avatar
1 vote

Integral of Hermite polynomial multiplied by $\exp(-x^2/2)$

Just wanted to add a quick and maybe more fun answer. Since Hermite functions are stable under Fourier transforms, we can write $$ I_n(\alpha) \equiv \int_{-\infty}^{\infty} \mathrm{d} x\, H_n(x) e^{-\...
Mert Okyay's user avatar
1 vote

If $f$ and $g$ coincide almost everywhere on $[a, b]$, then is $\int_a^b f(x) dx = \int_a^b g(x) dx$?

If $h=0$ almost everywhere, then it has a null integral. (To prove this, try showing it for characteristic functions, and then use the definition of the integral).
Maxime's user avatar
  • 319
1 vote

Double integral of the form exp(-a(x-y)^2)

I'm gonna sketch out the solution, the questions are welcome in the comments: First, apply the change of variables in the following form: $$ \left\{ \begin{array}{} u&=& x-y \\ v&=& x+...
Egor Larionov's user avatar

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