Linked Questions
32 questions linked to/from Definite integrals solvable using the Feynman Trick
4
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Methods to solve $\int _0^{\infty }\frac{x^{\frac{4}{5}}-x^{\frac{2}{3}}}{\ln \left(x\right)\left(x^2+1\right)}\:dx$
Find $$\int _0^{\infty }\frac{x^{\frac{4}{5}}-x^{\frac{2}{3}}}{\ln \left(x\right)\left(x^2+1\right)}\:dx.$$
I'd like to know in what ways can one approach this integral that can be found here, since ...
6
votes
2
answers
1k
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Integral $\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx$
Prove that
$$\int_{0}^{\frac{\pi}4} \ln(\sin{x}+\cos{x}+\sqrt{\sin{2x}})dx =\frac{\pi}{4} \ln2$$
I tried to use King's rule and to scale by $2$ and then to add the integrals, to get product of ...
5
votes
2
answers
240
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Does anyone have good logarithmic integrals? And logarithmic integral identities?
I have recently taken an interest in evaluating logarithmic integrals and would really love practice problems and especially, theorems, series expansions, and identities that have helped any of ya’ll ...
4
votes
2
answers
793
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Integral $\int_{0}^{1} \int_{0}^{1} \frac{1}{(1+x y) \ln (x y)} d x d y$
Evaluate $$\int_{0}^{1} \int_{0}^{1} \frac{1}{(1+x y) \ln (x y)} d x d y$$
I couldn't get very far on this one, so I would appreciate some help =)
My attempt so far (transcribed from the comments):
...
4
votes
2
answers
728
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$\int_0^\frac{\pi}{2}\frac{\ln(\sin(x))\ln(\cos(x))}{\tan(x)}dx$
I have the problem below:
$$\int_0^\frac{\pi}{2}\frac{\ln(\sin(x))\ln(\cos(x))}{\tan(x)}dx$$
I have tried $u=\ln(\sin(x))$ so $dx=\tan(x)du$
so the integral becomes:
$$\int_{-\infty}^0u\ln(\cos(x))du$...
3
votes
2
answers
383
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Integral $\int^{\infty}_0 \exp\left[-\left(4x+\frac{9}{x}\right)\right] \sqrt{x}\,dx$
How do I evaluate $$\displaystyle\int^{\infty}_0 \exp\left[-\left(4x+\dfrac{9}{x}\right)\right] \sqrt{x}\;dx?$$
To my knowledge the following integral should be related to the Gamma function.
I ...
3
votes
2
answers
515
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What methods can be used to solve $ \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $
I'm seeking methods to solve the following definite integral:
$$ I = \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $$
14
votes
1
answer
993
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A difficult double integral $\int_{0}^{1}\int_{0}^{1}\frac{x\ln x \ln y }{1-xy}\frac{dxdy}{\ln(xy)}$
How to evaluate $$\int_{0}^{1}\int_{0}^{1}\frac{x\ln x\ln y}{1-xy}\frac{dxdy}{\ln(xy)} ?$$
Any ideas on how to even start with this integral? It seems impossible to me.
There's a similar integral ...
9
votes
1
answer
684
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Evaluate $\int_0^{\pi/2} \frac{\arctan{\left(\frac{2\sin{x}}{2\cos{x}-1}\right)}\sin{\left(\frac{x}{2}\right)}}{\sqrt{\cos{x}}} \, \mathrm{d}x$
Evaluate: $$\int_0^{\frac{\pi}{2}} \frac{\arctan{\left(\frac{2\sin{x}}{2\cos{x}-1}\right)}\sin{\left(\frac{x}{2}\right)}}{\sqrt{\cos{x}}} \, \mathrm{d}x$$
I believe there is a "nice" closed ...
9
votes
1
answer
870
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Using Laplace transforms to evaluate$\int_{0}^{\infty}\frac{\sin^2(x)}{x^2(x^2 + 1)} dx$
Recently I've been playing around with Feynman's Trick to evaluate integrals. Obviously, one of it's many great features is that it allows derivatives to make expressions simpler. I was wondering ...
7
votes
1
answer
599
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Integral $\int_0^\infty \frac{x-\sin x}{x^3(x^2+4)} dx$
The following integral appeared on the $8$th Open Mathematical Olympiad of the Belarusian-Russian University. $$I=\int_0^\infty \frac{x-\sin x}{x^3(x^2+4)} dx$$ I used power series: $$x-\sin x = \sum_{...
3
votes
1
answer
247
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What's the common theme to look out for, while applying Feynman's Integral tricks? [duplicate]
I have seen Feynman's integral trick coming into use in many questions, but I don't really see a common way to recognize the format for it. Of course when a question comes with an explicit parameter, ...
2
votes
1
answer
265
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Evaluating $\int_0^t\frac{1}{\sqrt{x^3}} e^{- \frac{(a-bx)^2}{2x}} dx$
I've been trying to compute the following integral, but I havent been able to. Mathematica gives me an answer, but I would like to know how to get to that answer. For reference, this is the CDF of ...
2
votes
1
answer
196
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Leibniz integral rule for solving integrals
I am giving as homework to solve these integral
1) $\int \limits_{0}^{1} \frac{\arctan(y x)}{x \sqrt{1-x^2}}dx$
2)$\int \limits_{0}^{\frac{\pi}{2}} \frac{x}{\tan x}dx $ with the hint of $0\leq y \...
7
votes
0
answers
265
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Evaluation of $\int_0^1 \frac{\ln(1-x+x^2)}{x(1-x)}dx$
To add one more solution link to this answer listing Feynman's trick exercises, I'm posting herein a calculation by said technique of $I_{12}:=\int_0^1\frac{\ln(1-x+x^2)dx}{x(1-x)}$, viz.$$\begin{...