Linked Questions

4 votes
3 answers
190 views

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 ...
Dennis Orton's user avatar
  • 2,646
6 votes
2 answers
1k views

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 ...
Grentouce's user avatar
  • 301
5 votes
2 answers
240 views

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 ...
Person's user avatar
  • 1,113
4 votes
2 answers
793 views

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): ...
Flammable Maths's user avatar
4 votes
2 answers
728 views

$\int_0^\frac{\pi}{2}\frac{\ln(\sin(x))\ln(\cos(x))}{\tan(x)}dx$

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$...
Henry Lee's user avatar
  • 12.2k
3 votes
2 answers
383 views

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 ...
Max Wong's user avatar
  • 541
3 votes
2 answers
515 views

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: $$ I = \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $$
user avatar
14 votes
1 answer
993 views

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 ...
user avatar
9 votes
1 answer
684 views

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 ...
user avatar
9 votes
1 answer
870 views

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 ...
user avatar
7 votes
1 answer
599 views

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_{...
Zacky's user avatar
  • 27.2k
3 votes
1 answer
247 views

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, ...
Arsenic's user avatar
  • 321
2 votes
1 answer
265 views

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 ...
dleal's user avatar
  • 271
2 votes
1 answer
196 views

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 \...
Ahmad's user avatar
  • 2,358
7 votes
0 answers
265 views

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{...
J.G.'s user avatar
  • 116k

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