All Questions
36
questions
1
vote
3
answers
119
views
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 ...
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: $$\int \frac{2020x^{2019}+2019x^{2018}+2018x^{2017}}{x^{4044}+2x^{4043}+3x^{4042}+2x^{4041}+x^{4040}+1}...
- 6,620
5
votes
1
answer
141
views
Evaluate $\int_{0}^{\pi/2}x\ln\left(\tan x\right)\,dx$ [duplicate]
Evaluate $$\int_{0}^{\pi/2}x\ln\left(\tan x\right)\,dx$$
First we will work out the complex integral of the function $$\displaystyle{f\left( z \right) = \frac{{{z^2}}}{{{e^z} - 1}},{\text{ }}z \ne 0}...
user1235604
5
votes
2
answers
174
views
Double integral $\int_{0}^{1}\int_{0}^{1}\frac{x^{a-1}y^{b-1}}{(1+xy)\ln(xy)}\,dx\,dy$
Double integral $$\int_{0}^{1}\int_{0}^{1}\frac{x^{a-1}y^{b-1}}{(1+xy)\ln(xy)}\,dx\,dy$$
$\displaystyle{a,b>0}$
my work
$$I\left( {a,b} \right) = \int\limits_0^1 {\int\limits_0^1 {\frac{{{x^a}{y^...
user1235604
6
votes
4
answers
241
views
Evaluate $\int_{0}^{1}\{1/x\}^2\,dx$
Evaluate
$$\displaystyle{\int_{0}^{1}\{1/x\}^2\,dx}$$
Where {•} is fractional part
My work
$$\displaystyle{\int\limits_0^1 {{{\left\{ {\frac{1}{x}} \right\}}^2}dx} = \sum\limits_{n = 1}^\infty {\...
user1235604
2
votes
1
answer
140
views
Explicit expression for integral of $(1-a~x^{-1/2}-b~x)^{-3/2}$
I wonder if there exists an explicit solution to the integral $$\int_{1}^{x} \frac{d{u}}{\sqrt{(1-a/\sqrt{u}-b~u)^{3}}}\tag{1}$$ Any help or advice is welcome.
Appendix
The full expression with ...
- 175
0
votes
1
answer
50
views
How to approach this integral? $ \int_0^\Theta {\frac{1}{a+b\cot\left({\pi{x}^c}\right)}} dx $
Could someone suggest how to approach this integral:
$$
\int_0^\Theta {\frac{1}{a+b\cot\left({\pi{x}^c}\right)}} dx
$$
where $a$, $b$, $c$ are scaler values, and $0<\Theta<1$. I have tried some ...
2
votes
1
answer
79
views
Integral $ I = \int_{-\infty}^{\infty} x^{T}A x \phantom{\times} \textrm{exp} \left( x^{T}Bx-\ln{x^{T}x} \right) dx, $
I need help with this multiple integral
$$
I = \int_{-\infty}^{\infty}
x^{T}A x \phantom{\times} \textrm{exp} \left( x^{T}Bx-\ln{x^{T}x} \right) dx,
$$
where $x\in\mathbb{R}^n$, $A\in\mathbb{R}^{n\...
- 21
6
votes
1
answer
275
views
Closed form for $\int \sqrt[n]{\tan x}\ dx$
I was solving $\displaystyle\int\sqrt[n]{\tan x}\ dx$.
Here's my method:
$$\begin{align}\int\sqrt[n]{\tan x}\ dx &= \int\frac{n \cdot t^n}{1 + (t)^{2n}}\tag{1}\ dt\\& = n \int\sum_{k=0}^\infty ...
- 1,614
0
votes
2
answers
93
views
Algorithm for integration of rational functions in t, $\sqrt{p(t)}$, where $p(t) = at^2 + bt + c$
Let $p(t) = at^2 + bt + c \in \mathbb{R}[x]$ be a degree $2$ polynomial.
Let $r(x, y) = \frac{a(x, y)}{b(x, y)} \in \mathbb{R}[x, y]$ be a rational function in $x$ and $y$.
I am interested in a closed ...
user900250
1
vote
0
answers
97
views
Is there an antiderivative for $e^{-\left( x + \frac{1}{x}\right)}$?
I was playing around with some integrals I made up myself and was trying to find a closed-form for
$$
\int_{0}^{t} e^{-\left( x + \frac{1}{x}\right)} \ dx, \qquad t < \infty
$$
I'm aware that if ...
- 7,273
0
votes
1
answer
59
views
General integration rules for integrals of simple products and quotients of $x$, $f(x)$, $f'(x)$ and higher derivatives of $f$?
My question is about symbolic integration.
Which simple general integration rules valid for all integrable complex-valued functions $f$ of one complex variable are there for the integrals that are ...
- 7,212
2
votes
1
answer
470
views
Equivalent of product rule for integrals: $\int f'(x)g'(x)dx$
Recently I've been working on some integration problems and I have come across some integrands of the form $f'(x)g'(x)$. I've found myself wondering; I know the product rule for differentiation, ...
- 6,560
2
votes
0
answers
166
views
Is $\int \exp(x \exp(x)) dx$ a well-known integral?
Recently puzzling around with the Lambert-$W$ function, I have become intrigued by the integrals:
$$\int e^{xe^{x}}dx \tag{1}$$
$$\int e^{xb^{x}}dx \tag{2}$$
$$\int e^{axb^{x}}dx \tag{3}$$
the latter ...
- 909
-3
votes
1
answer
128
views
A solution for the integral $\int \frac{\sqrt[3]{x} }{\sqrt[3]{x^2} -\sqrt{x} } dx $ and WA's "invalid input".
Problem:
$$\displaystyle \int \frac{\sqrt[3]{x}}{\sqrt[3]{x^2} -\sqrt{x} } dx$$
My attempts:
$$\displaystyle \int \frac{\sqrt[3]{x}}{\sqrt[3]{x^2} -\sqrt{x} } dx={\displaystyle\int} \dfrac{1}{6x^\...
user548054