All Questions
Tagged with calculus-and-analysis trigonometry
62 questions
0
votes
1
answer
61
views
Computing an exact range with FunctionRange
When using FunctionRange to compute the range of the two-argument ArcTan function, Mathematica fails to return an exact answer.
...
CommunityBot
- 1
1
vote
3
answers
229
views
Symbolic integral over real functions with interger parametres evaluates to complex numbers
I want to compute some quantum mechanical matrix elements. I use that:
...
- 29.1k
1
vote
0
answers
64
views
How can I speed up symbolic integration of this highly oscillatory function?
I am trying to figure out how to speed up this symbolic integration. I have tried parallelization and numerical integration (both I am failing to implement).
Here is the function:
...
- 5,820
1
vote
2
answers
174
views
Confused about the output of `CosIntegral`
I am interested in the properties of the cosine integral function, in this case the anti-derivative of Cos[Pi*x]/x, which Mathematica evaluates to ...
- 245k
1
vote
0
answers
75
views
How can I ask Mathematica to give the explicit real form of the given function?
In part of my calculations, I obtain this expression which contains the imaginary unit I but I expect that this expression might be real (from the comments below, ...
- 195
2
votes
3
answers
164
views
Evaluate using Mathematica or otherwise $\int_{0}^{\pi/2}\sin^n x \ T_n(\sin x)\ dx$
Denote $T_n(x)$ as Chebyshev polynomial of the first kind (see here). Then I need to evaluate for $n$ a odd natural number $$\int_{0}^{\pi/2}\sin^n x \ T_n(\sin x)\ dx $$
I am requesting a code with ...
- 4,628
3
votes
2
answers
244
views
Is there an option to automatically make solve return the principal value of inverse trig functions?
I am porting some of my Maple packages to Mathematica. In Maple, solve(sin(y)=x,y) returns arcsin(x) and I'd like to get same ...
- 29.1k
5
votes
4
answers
702
views
How to calculate the integral only in real domain?
I have the following integral:
Integrate[1/(Sqrt[2] + Cos[4 X] + Sin[4 X]), X,
Assumptions -> Element[X, Reals]] // Simplify
...
- 13.1k
13
votes
1
answer
390
views
Series with ArcTan gives wrong symbolic answer in Wolfram Language
Bug introduced after 9 and persisting through 13.1. Resolved in 13.2
Recently, I have found a very bad problem with Wolfram Language. It gives the wrong answer for a quite simple expression!
When ...
- 17k
2
votes
1
answer
103
views
How can I make sure that the two given numbers are exactly the same?
I have the given numbers $n1$ and $n2$. How can I make sure that these two numbers are exactly the same? Numerically, using {N[n1,40], N[n2,40]}, by increasing the ...
- 6,027
1
vote
2
answers
105
views
How to ask Mathematica to rewrite the larger arguments of sine function as a smaller number which is multiple of $\frac{2\pi a}{11}$
I have a trigonometric function $Exp$ where $\{f,g,h,k\}$ are some parametric functions, and $a\in\mathbb{N}$ and $x>0$.
$$ Exp=f \cdot \sin \left(\frac{58 \pi a}{11}+x\right)+g\cdot \sin \left(\...
- 6,027
2
votes
1
answer
121
views
Slow integration of trigonometric expression speeds up after TrigReduce
I'm trying to integrate a trigonometric expression as
Integrate[Cos[2 t] Cos[3 t] Cos[4 t] Cos[5 t] Sin[2 t] Sin[3 t], {t, 0, 2 Pi}]
The final results is correct (...
- 245k
1
vote
0
answers
49
views
Power of trigonometric function matrix [closed]
Calculate the m-power of the following trigonometric function matrix (m>0):
$\boldsymbol{A}=\left(\begin{array}{cc}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi\end{array}\...
- 2,425
3
votes
0
answers
140
views
Cannot Understand nth Derivative of x/ArcTan[x]
The nth derivative of x/ArcTan[x]:
f[x_, n_] = D[x/ArcTan[x], {x, n}]
Evaluates to:
I cannot get this general from to return ...
0
votes
0
answers
108
views
Simplification of long trigonometric expression taking a long time
I have tried to Simplify as well as FullSimplify a long trigonometric expression to get a simplified form of the expression but it takes a very long time almost hours and sometimes it also shows error ...
- 135
4
votes
1
answer
293
views
Mathematica vs Rubi --- integrate $f(x,y) = x \sin^2 x + a x y$ --- Mathematica got this round? [closed]
Consider the function:
$$
\begin{equation}
f(x,y) = x \sin^2 x + a x y
\end{equation}
$$
with $a$ being a constant.
We want to do the indefinite integral over $x$ and then over $y$. We do that using <...
- 151k
1
vote
0
answers
39
views
How to sort out only terms that relates to certain variable in the results of TrigReduce? [closed]
I have a large function $F(\theta,\alpha_1,\alpha_2,...)$ which involves only trigonometric polynomial of these angles. I want to calculate $\frac{1}{2\pi}\int_0^{2\pi} Fd\theta$, but the function is ...
- 111
0
votes
1
answer
152
views
NIntegrate giving different results after a change of variables
When I compute numerically the following double integral:
\begin{equation}
f(x,y)=\int_{-\infty}^{+\infty}da\int_{-\infty}^{+\infty}db\, \cosh\left[\frac{\sinh^{-1}(a)-\sinh^{-1}(b)}{2}\right]\, \exp\...
- 56.9k
3
votes
1
answer
275
views
Why can I not numerically integrate a smooth, integrable function?
Consider the function $\csc(t)\sin(3t)$ on the interval $[0,\pi]$. This is a perfectly well-behaved smooth function, as can be seen when plotting it:
...
2
votes
1
answer
204
views
DSolve fails to find solution based on elementary function
I have two differential equations for which DSolve does not find a solution, however, there are simple solutions in terms of elementary functions, and more ...
- 151k
7
votes
4
answers
1k
views
How to transform the result of this integral into powers of $\sin\,x$?
I want to calculate the integral
$I = \displaystyle\int{\cos ^5 x\, \mathrm dx}$. Using the variable $t = \sin x$, I get
$$I = \sin x - \frac{2}{3}\sin^3 x + \frac{1}{5}\sin^5 x + C.$$
With ...
1
vote
0
answers
75
views
How to get a result for $\int{\sqrt{a^2 \left(\sin{a x}-2 \sin{2 a x}\right)^2+1} \, \mathrm{d}x}$?
I want to obtain the result of this integral where $\mathbb{R}\ni a,x>0$
$$\int{\sqrt{a^2 \left(\sin{a x}-2 \sin{2 a x}\right)^2+1} \, \mathrm{d}x}$$
but Mathematica does not give an answer. Of ...
- 904
0
votes
2
answers
117
views
How to extend this integration to all the reals?
Doing an exercise on calculus with 12.2 on Windows 10 Pro, I obtain
a = Integrate[Sqrt[1 + Sin[t]], {t, 0, x}, Assumptions -> x \[Element] Reals]
...
- 56.9k
0
votes
1
answer
204
views
Derivatives of trigonometric functions
Let's use the following sample case
Clear["Global`*"];
t1 = ArcTan[y/(x - x1)];
f = (3*(Cos[t1])^2 - 1);
der = D[f, x]
which gives
...
- 40k
1
vote
1
answer
148
views
Symbolic Integration with trigonometric functions
I have this trigonometric integral which takes forever to evaluate
$\int dx \frac{4 \left(\cos \left(p_1 x\right)-\frac{\sin \left(p_1 x\right)}{p_1 x}\right) \left(\cos \left(p_2 x\right)-\frac{\...
- 245k
5
votes
4
answers
466
views
NIntegrate Oscillating kernel
I have a finite highly oscillating integral of the type:
NIntegrate[x^2 Exp[I f[x] s] , {x, 0, 1}]
where f is a simple function ...
- 1,235
0
votes
1
answer
51
views
$\alpha \int_{0}^T \text{Sin}(fi−fk+Δf)∗t) \, dt$ — Why is it not giving wrong answer? [closed]
$$\alpha \int_{0}^T \text{Sin}(fi−fk+Δf)∗t) \, dt $$
I am integrating this in Mathematica,
My code is
(1/2)*α*Integrate[Sin[2*Pi*(fi - fk + δf)*t], {t, 0, T}] + Ν1
...
- 29
0
votes
1
answer
229
views
Simple Question: How to simplify what is inside the Sin function like below?
I have this code of Mathematica
(1/(2 (fj - fk) π))
Cos[(fj - fk) π T + (fj - fk) π (T + δt)] Sin[(fj -
fk) π T - (fj - fk) π (T + δt)]
There are a few ...
- 12.8k
3
votes
1
answer
92
views
Establish the equivalence of two inverse trigonometric function based formulas
In a comment to
3D5D
user JimB provided an answer to the question posed there of finding the "two-quater[nionic]bit" absolute separability Hilbert-Schmidt probability. (Previously, in ...
- 19.8k
2
votes
1
answer
143
views
Mathematica outputs a trigonometric integral ($\sec^3$) in a form I can't prove
The indefinite integral is of course $1/2 ( \sec(x) \tan(x) + \ln | \sec(x) + \tan(x) | ( + C)$.
Mathematica gives:
...
3
votes
1
answer
162
views
Could anyone help me find the EXACT value?
Could anyone please help me find the EXACT value (not numerical value) of this by Mathematica or by mathematical reasoning? Thanks a lot.
...
CommunityBot
- 1
2
votes
0
answers
58
views
Why Integrate does not obtain this conditional result automatically for orthogonality of cosines?
V 12.1 on windows.
I remember asking something similar many years ago. I was hoping Mathematica now could have done this automatically:
$$
\int_{-\pi}^{\pi} \cos (n x) \cos (m x) \, dx
$$
For ...
- 151k
6
votes
4
answers
993
views
Evaluate the defining Integral of the Bessel functions of the first kind
I am trying to evaluate the integrals
$$ \int\limits_{-\pi}^{\pi} \mathrm{e}^{\mathrm{i}(x\sin t - nt)} \mathrm{d}t $$
and
$$ \int\limits_{-\pi}^{\pi} \mathrm{e}^{\mathrm{i}x\sin t} \mathrm{d}t $$
...
5
votes
2
answers
132
views
Primitive of a continuous function over $\Bbb R$
A continuous function over $\Bbb R$ has a primitive that is also continuous over $\Bbb R$.
However, it often happens with Mathematica and other CAS that the result is not continuous, especially with ...
- 245k
12
votes
7
answers
4k
views
Trying to prove that $x\sin(\frac{\pi}{x})\ge\pi \cos(\frac{\pi}{x})$ for $x\ge 1$
As an intermediate step, consider the function:
f[x_] := x Sin[Pi/x]
I want to prove that this function is increasing for $x\ge 1$. This can be done with the ...
7
votes
1
answer
1k
views
Integral that yields ArcTan function rather than ArcSin
When I enter the following Integral problem to Mathematica,
$$
\int \frac{1}{\sqrt{a^2-x^2}}\textrm{d}x
$$
The answer yielded by Mathematica is
$$
\arctan\left({\frac{x}{\sqrt{a^2-x^2}}}\right)
$$
...
0
votes
0
answers
142
views
Minimal Surface of Revolution - Integrate Challenge
I'm trying to use Mathematica to go from equation (11) to equation (12) in this example of a Minimal Surface of Revolution. There is a MMA notebook on that link, but it only shows how to find the ...
0
votes
0
answers
55
views
Different integration results when TrigToExp used
I am trying to do a double integral over a domain $x \in (\theta_1,\theta_2), y \in (0,\theta_1)$. Instead of doing a definite integral I am doing an indefinite integral and taking limits. All that is ...
- 183
0
votes
2
answers
181
views
How to get the numerator of this fraction involving `Csc` or `Sec`?
I would like to avoid multiplying explicitely by the denominator, because I'm working with huge fractions.
Consider
...
- 132k
4
votes
2
answers
691
views
Mathematica gives an unexpected answer for Integrate [closed]
I need to integrate the following:
\begin{equation}\tag{1}
\frac{\sqrt{C + (1 - C) x^3}}{x},
\end{equation}
where $0 < C < 1$ and $x$ is a positive variable (then $x^3 \ge 0$). When I integrate:...
- 245k
4
votes
0
answers
165
views
Inconsistent results for symbolic trigonometric integral
I am trying to evaluate (on Mathematica v.9) the integral
\begin{equation}
\int_0^x [\sin (x) \sin (2 y)-\sin (2 x) \sin (y)]^t \,\mathrm{d}y,
\end{equation}
where $t$ is an even, positive integer. ...
1
vote
1
answer
86
views
Plot of integral of summed `Sinc` series is incorrect
I have curve given by summing a small, finite series of Sinc functions, and I want to plot both the curve and its integral. In principle, it's easy:
...
- 2,335
3
votes
2
answers
191
views
Indefinite ArcTan integral does not return the (known) real expression but returns complex expression
I have a problem with an indefinite integral which I solved using MMA in back in 2010 (must have been version 7 or 8) successfullly and it gave an elegant result. However, if I run it with MMA 11.3 (...
- 113k
0
votes
0
answers
141
views
User defined ArcTan function
I am trying to integrate an equation that requires an angle term calculated using ArcTan[x,y]. A number of issues have been mentioned about ...
- 371
0
votes
2
answers
140
views
Bug in limit evaluation? [closed]
When I try to evaluate the limit
$$\lim_{x\to\infty}x\sin(2\pi e x!)$$
Mathematica yields Indeterminate as an answer, however, the solution is known to be $2\pi$. What could be the cause of this ...
- 6,133
1
vote
1
answer
187
views
Integrals for trigonometric functions
I was able to obtain some indefinite integrals for the trigonometric function manually,
$\Gamma^{pq}_n = \int \frac {\cos^p x \sin^q x} {(1+A \cos x)^n} dx$
for $(p,q,n)\geq 0$ in recursive form. ...
- 245k
0
votes
1
answer
187
views
Indefinite Integral for rational trigonometric functions without hypergeometric functions
How to find indefinite integrals for the rational trigonometric functions of the form
$\qquad\Lambda^{pq}_m = \int \frac{\cos^p x\sin^q x}{(a+b\sin x +c\cos x)^m} dx$
where $(p,q,m)>0$ using ...
- 8,448
19
votes
1
answer
852
views
Speeding up trigonometric integral
Context
On a possible non trivial toric topology for the Universe (nothing less!).
Problem
I would like to carry out the following integral for $\ell=2,4\cdots 20$.
$$\int _0^{\pi }\int _0^{2 \pi }...
0
votes
1
answer
257
views
Arduous trigonometric simplification task
I have equations that are products of sines and cosines that I want to integrate. As an example, the integrand may look something like this:
$-\cos \left(\frac{\sqrt{3} \pi x (p+q)}{A}\right) \sin \...
- 17.4k
1
vote
2
answers
2k
views
Find the explicit solution of the integral
I want to calculate the integral:
$$A_n=\int_0^{2\pi}\cos^{2n}(\theta-\theta_0)\mathrm d\theta$$
assuming that we are familiar with the relation:
$$\cos^{2n}x=\frac1{2^{2n}}\binom{2n}{n}+\frac1{...