All Questions
519
questions
1
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3
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Is there any nonlinear function $f(x)$ for which both $\sin (f(x))$ and $\cos(f(x))$ have closed-form integrals?
I'm interested in closed-form antiderivaties of elementary functions.
The question arises from seeing that something as simple as $\int\sin(x^2)dx$ and $\int\cos(x^2)dx$ generates the Fresnel ...
1
vote
2
answers
147
views
Is there a closed form for the integral $\int_{0}^{\pi/4}x^{k}\ln\tan x\, dx$, where $k$ is a natural number?
To start, I am aware that our integral $I(k)=\int_{0}^{\pi/4}x^{k}\ln\tan x\, dx$ is equal to $$I(k)=\int_{0}^{\pi/4}x^{k}\ln\sin x\, dx-\int_{0}^{\pi/4}x^{k}\ln\cos x\, dx$$, but I cannot seem to ...
0
votes
2
answers
140
views
How to evaluate $\int_{0}^{1} \int_{0}^{1} \frac{4y \, da \, dy}{(y^2 + ay + 1)(y^2 - ay + 1)}$? [duplicate]
Question: How to evaluate $$\int_{0}^{\pi/2} \ln \left( \frac{2 + \sin x}{2 - \sin x} \right) \, dx$$
My attempt
The original integral is:
$$
J = \int_{0}^{\pi/2} \ln \left( \frac{2 + \sin x}{2 - \...
8
votes
1
answer
235
views
How to solve $\int \frac{2020x^{2019}+2019x^{2018}+2018x^{2017}}{x^{4044}+2x^{4043}+3x^{4042}+2x^{4041}+x^{4040}+1}dx$
One of my friends sent me a list of integrals (all without solutions ) one of those problems is: $$\int \frac{2020x^{2019}+2019x^{2018}+2018x^{2017}}{x^{4044}+2x^{4043}+3x^{4042}+2x^{4041}+x^{4040}+1}...
8
votes
3
answers
264
views
How to evaluate $\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, dx$?
Question: How to evaluate $$\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, dx?$$
My attempt
We will denote the main integral as $\Omega$.
$$\Omega=\int_0^{\frac{\pi}{4}} \tan(x) \ln^2(\sin(4x)) \, ...
5
votes
2
answers
239
views
How can I show that $\int_0^{\frac{\pi}{2}}\sin\left(\frac{x}{2}\right) \text{arctanh}\left(\sin(2x)\right) \ dx$.
Question: How can I show that
\begin{align}
& \int_0^{\frac{\pi}{2}} \sin\left(\frac{x}{2}\right) \text{arctanh}\left(\sin(2x)\right)\,dx
\\[2mm] = & \
{\small\log\left(\left(2\sqrt{2-\sqrt{2}}...
7
votes
2
answers
114
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How to Evaluate the Integral $\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{1+\sin(y)}\ln(\sin(y))}{\cos(y)}dy?$
Question: How to Evaluate the Integral $$\int_{0}^{\frac{\pi}{2}}\frac{\sqrt{1+\sin(y)}\ln(\sin(y))}{\cos(y)}dy?$$
My attempt
I'm looking for a method to evaluate it. I've attempted a substitution to ...
3
votes
1
answer
146
views
how to evaluate this integral $\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$
Question statement: how to evaluate this integral $$\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$$
I don't know if there is a closed form for this integral or not.
Here is my attempt to solve ...
8
votes
4
answers
426
views
how to evaluate $\int_0^1{\ln ^3\left( 1-x \right) \ln ^2\left( 1+x \right) \text{d}x}$
Integral: how to evaluate $$\int_0^1{\ln ^3\left( 1-x \right) \ln ^2\left( 1+x \right) \text{d}x}$$
Same context
I'm not sure of the closed form of the integral, as I haven't evaluated it yet. ...
7
votes
3
answers
431
views
How to evaluate $\int_0^1 \ln ^3(1+x) \ln (1-x) d x$?
QUESTION:How to evaluate $$\int_0^1 \ln ^3(1+x) \ln (1-x) d x$$?
I'm not sure of the closed form of the integral, as I haven't evaluated it yet. However, after evaluating the integral $$\int_0^1 \ln (...
1
vote
0
answers
66
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How to integrate $\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$ [duplicate]
Question; how to integrate $$\int_0^\frac{1}{2}\frac{\ln(1+x)}{x}\ln\left(\frac{1}{x}-1\right)\mathrm{d}x$$
here is my attempt to solve the integral
\begin{align} I&=\int_0^\frac{1}{2}\frac{\ln(1+...
9
votes
1
answer
384
views
evaluation of $\sum_{n=1}^{\infty} (-1)^{n-1} \frac{H_{n} H_{n+1}^{(2)}}{(n+1)^{2}}$ and other Euler sums
I was trying to evaluate this famous integral $$\int_{0}^{1} \frac{\ln (x) \ln^{2}(1+x) \ln(1-x)}{x} \ dx $$
Here is my attempt so solve the integral
\begin{align}
&\int_{0}^{1} \frac{\ln (x) \ln^{...
7
votes
1
answer
295
views
how to evaluate $\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$
Question: how to evaluate $$\int_0^{\infty} \frac{x \ln ^2\left(1-e^{-2 \pi x}\right)}{e^{\frac{\pi x}{2}}+1} d x$$
MY try to evaluate the integral
$$
\begin{aligned}
& I=\int_0^{\infty} \frac{x \...
7
votes
3
answers
228
views
Closed form for $\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$?
Is there a closed form for $I=\int_0^{\pi/2}\arctan\left(\frac12\sin x\right)\mathrm dx$ ?
Context
Earlier I asked "Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\...
4
votes
3
answers
220
views
Show that $\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$
Problem: Show that $$\int_{0}^{1} \frac{\tan^{-1}(x^2)}{\sqrt{1 - x^2}} \, dx = \frac{1}{2}\pi \tan^{-1}\left(\sqrt{\frac{1}{\sqrt{2}} - \frac{1}{2}}\right)$$
Some thinking before trying
At least we ...