Tagged Questions

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How to evaluate a definite integral involving the product of two sines?

The input signal for a given electronic circuit is a function of time $V_{in}(t)$. The output signal is given by $$V_{out}(t) = \int_0^t \sin(t-s) V_{in}(s)\,ds$$ Find $V_{out}(t)$ if ...
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How to integrate $\sqrt{1+(2/3)x}$?

How would you solve the following (step by step please!): $$\int^6_5\sqrt{1+\frac23x}\ dx$$ I started with $u=1+\frac23x$, $du=\frac23\,dx$, now what?
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Improper integral of rational function $k^2/(1+a^2k^2)^2$

I've got the integral $\int^\infty_{-\infty} dk \frac{k^2}{(1+a^2 k^2)^2}$ where $a$ is a real number. I can't seem to find a $u$-substitution or trigonometric substitution that will work. Any ...
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Volume of Solid Revolution

region bounded by $$y=x$$ and $$y=x^2$$ a) find the volume of the solid of revolution formed by revolving R region about the line $x=2$. b) find the volume of the solid of revolution formed by ...
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If a function $f$ is bounded in $[a,b]$ and if for each $x\in(a,b)$, $f$ is integrable over $[x,b]$, Then $f$ is integrable over $[a,b]$

I got this question which popped up in my mind: Prove or disprove: If a function $f$ is bounded in a closed interval $[a,b]$ and if for each $x\in(a,b)$, $f$ is integrable over $[x,b]$, Then $f$ is ...
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Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$

Here is another integral I'm trying to evaluate: $$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx.\tag1$$ A numeric approximation is: ...
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CAS Can't Solve the Integral

I am trying to solve this integral; $$\int^a_{-a}\frac{E[\frac{-(4b_1 b_2)}{((x+z)^2+(b_1-b_2)^2 )}]}{((x+z)^2+(b_1+b_2)^2)\sqrt{((x+z)^2+(b_1-b_2)^2 )}}dx$$ E is the complete elliptic integral of ...
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Evaluating the integral $\int_{-1}^{1}\frac{x^2\,dx}{e^x+1}$

Can this integral be evaluated properly by using highschool integration skills ? $$\int_{-1}^{1}\dfrac{x^2}{e^x+1}\,dx$$ (Original image at http://i.stack.imgur.com/AqOsX.png) Judging from what I ...
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A logarithmic integral

How can I evaluate following logarithmic integral: $$\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$$
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Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

I want to find a closed form for this integral: $$I=\int_0^1\frac{\ln^2x}{\sqrt{x^2-x+1}}dx\tag1$$ Mathematica and Maple cannot evaluate it directly, and I was not able to find it in tables. A numeric ...
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Question about the limit definitions of derivative and definite integral

Actually this question may look simple and basic, but it is about something which bugs me for a long time, since when I took my first calculus classes. The limit $\lim_{x \to a}f(x) = L$ is defined ...
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Evaluate $\int_{0}^{1} \ln(x)\ln(1-x)\,dx$

Evaluate the integral, $$\int_{0}^{1} \ln(x)\ln(1-x)\,dx$$ I solved this problem, by writing power series and then calculating the series and found the answer to be $2 -\zeta(2)$, but I don't ...
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Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$

A few days ago, I posted the following problems Prove that \int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}\\[20pt] -\int_0^{\pi/2}\ln^3(\cos ...
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Why is $\int_0^{2\pi} e^{i\,k\rho[\sin\alpha\cos\alpha-\sin\theta\cos(\phi-\beta)]}\mathrm{d}\beta = 2\pi J_0(k\rho\xi)$?

The following is an integral in Jackson Classical Electrodynamics (3rd ed.). In equation (10.112) the integral  \int_0^{2\pi} ...
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Prove that $\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$

Prove that $$\int_0^{\pi/2}\ln^2(\cos x)\,dx=\frac{\pi}{2}\ln^2 2+\frac{\pi^3}{24}$$ I tried to use by parts method and ended with \int \ln^2(\cos ...
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Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2$

How to prove$$\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln(\cos x))^2 \right) \, dx=\pi\ln\ln2$$ I don't know how to answer it. When I asked this integral to my brother, ...