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Questions tagged [calculus-and-analysis]

Questions related to the calculus and analysis branches of Mathematica, including, but not limited to, limits, derivatives, integrals, series, and residues.

2
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1answer
48 views

Singular Integration

I am trying to simplify the following integral but getting no answer. Any help in how to get the resulting function as a function of t would be much appreciated? ...
1
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0answers
47 views

Getting Mathematica to supply derivatives in standard format [duplicate]

I will like to use the results of Mathematica outputs in formatted text for students. However, I do not know how to get rid of the variable arguments and the use of raised indices to indicate ...
1
vote
1answer
70 views

An optimization problem over two variables

I have the function fun[x_,y_, t_] = Cos[x] + Exp[I y] Sin[x] Sin[t]; such that $0\le {\rm x}\le \pi$, $0\le {\rm y}\le 2\pi$, and $0\le {\rm t} \le 10$. I compute ...
0
votes
5answers
47 views

Expressing D[1/D1, x] in terms of D1 when D1 = 1 + x

For simplicity, taking D1 to be equal to 1 + x, I want to evaluate the derivative of 1/D1 (...
0
votes
1answer
41 views

Integrating in polar differs from simple integration [on hold]

I have same line in orthogonal and polar coordinates, it is Cos[t] in first quadrant. But when I integrate those, result differs and I dont understand why: ...
2
votes
0answers
48 views

Issues with symbolic integration

I am trying to do the following indefinite integral, ...
0
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0answers
58 views

How to Solve This Integral?

Mathematica insists that the below integral does not converge, though it has been solved by my peers, so I am inclined to think that it does. q is an arbitrary variable. ...
0
votes
1answer
22 views

Limit function and DirectedInfinity as output: Interpretation

I tried to compute the following limit: In[12]:= Limit[-A/2 x - 2 x, x -> +Infinity] Out[12]= DirectedInfinity[-2 - A/2] Does the output tell me that the ...
-4
votes
0answers
69 views

Mathematica evaluates D[x, Log[x]] as 0 without raising any error [closed]

I thought the answer should be $ \frac{\partial x}{\partial \ln{x}} = \frac{1}{\frac{\partial \ln{x}}{\partial x}} = x $ Why Mathematica 11.3 does not know that it made a mistake? One possible ...
0
votes
1answer
78 views

DiracDelta is seldom chosen as the answer [closed]

It is well known that Limit[ ϵ/π/(x^2 + ϵ^2), ϵ -> 0] is the Dirac Delta function DiracDelta[x] Yet Mathematica gives ...
1
vote
1answer
33 views

Using rule with derivative

Assuming I have an expression like: D[a[t, r], r] + D[D[a[t, r], r],r] + D[D[a[t, r], t],r] I would like to change variables, by expression everything in terms of ...
1
vote
1answer
59 views
7
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2answers
741 views

Integral with DiracDelta. Can Mathematica be made to solve this?

This was an exam question. $$ \int_0^{2 \pi} \delta(\sin^2(\theta) -x ) \,d\theta $$ Direct use of Integrate on it does not give the solution. Is there a ...
2
votes
3answers
164 views

Mathematica won't evaluate this integral

The function (in raw input form) and bounds I used are as follows: ...
0
votes
2answers
97 views

Difficult limit [on hold]

I try to calculate the following limit to calculate the constants C[1] and C[2]: ...
0
votes
0answers
42 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 ...
0
votes
1answer
46 views

How do I compute the infinity-norm of a function? [closed]

I have a technical question on Mathematica 11. I have to compute the $\infty$ norm of q function, $f(x,y)$, on a rectangular domain $[a, b] \times [c,d]$. How can I do this with Mathematica?
3
votes
2answers
442 views

Speed up this NIntegrate

Is there any trick to speed up this numerical integral: ...
1
vote
1answer
42 views

How does one test whether or not an integration has succeeded?

This must be a very simple matter--but how to address it eludes me at the moment. I have a series of integrands--and want to flag whether the symbolic integation has succeeded or not (in order to ...
0
votes
3answers
108 views

Limit of this definite integral [closed]

How can I compute this integral for $y\rightarrow0$ ? $$ \int_0^1{\frac{y(1-x)^2(1+x)}{x+(1-x^2)y} dx} $$
0
votes
1answer
96 views

Numerical integration giving trouble

I am trying to do the following integral numerically, $$\rho(\theta_{j},\phi_{j})=\int\frac{\sin{\theta_{i}} d\theta_{i} d\phi_{i}}{\sqrt{2+2[\cos(\theta_{i})\cos(\theta_{j})+\sin{\theta_{i}\sin{\...
2
votes
2answers
623 views

Weird result in complex limit

I am trying to evaluate a limit: gamma[w_] = Sqrt[-(u*e)w^2 + I*(u*s)w]; Limit[Re[gamma[x]], {x -> DirectedInfinity[1]}] I calculated the limit by hand, and ...
1
vote
1answer
28 views

Verify that a PDE is satisfied by a matrix consisting of functions satisfying a different PDE

My problem is as follows: I have two matrices $P$ and $Q$ which are made up of algebraic expressions of a function $q(x,t)$ which solves a somewhat complicated pde, and its derivatives, for example, ...
0
votes
2answers
76 views

Integration gives a complex valued result

There are many similar questions but they all seem to be integrating functions that are more involved than just a product of polynomials and the issue has more to do with the integrand than the ...
1
vote
1answer
55 views

How can I solve this integration or does it have a closed form solution?

The integral I am dealing with is below. I need to find the closed-form expression of this integral. $\int_0^\infty \ln(1+\frac{A}{1+B+Cx})\frac{e^{-x/M}}{M}dx$ Here, $A$, $B$, $C$ and $M$ are ...
1
vote
1answer
67 views

Centroid of a cosine shaped rod

There's a problem in Hibbeler's 14th edition of Statics for Engineering that I've tried a few times and couldn't solve. The problem is shown below. It wants me to find the centroid coordinates of a ...
10
votes
2answers
803 views

Why does Arg'[1. + I] return -0.5?

From the document we know that Arg[z] gives the gives the argument of the complex number z. Then how about ...
3
votes
3answers
189 views

Approximating integral with small parameter

I want to approximately compute integral $$I =\int_0^1 dx \frac{x(2-x)(1-x)}{(1-x)^2+\mu x}$$ assuming that $\mu$ is small. I tried ...
0
votes
1answer
45 views

integration of discontinuous function [closed]

I am sure this is an easy question. Although I am new in mathematica. I want to integrate a function that has some (removable) singularities. $ f_h(x) = -x\cot{h\pi x} - \frac{1}{\pi h}\sum_{j=1}^{h} ...
0
votes
0answers
35 views

Calculate a definite integral containing an ExpIntegral

I need to calculate the following integral Integrate[Exp[-(y/bn)] (-y)^c ExpIntegralE[d, p y], {y, 0, z/x}] Where $z<0,z\in \mathbb{R}$, $x\in\mathbb{R},x>...
1
vote
1answer
45 views

Closed form solutions to functions involving Erfc

Is there a better way to get a closed form solution in terms of sigma? I've already attempted a significant amount of simplification up to this point and am unsure ...
0
votes
1answer
65 views

Solve a constrained four-dimensional integration problem for a “magic simplex” probability

Equation (7) in the 2012 paper, "Complementarity Reveals Bound Entanglement of Two Twisted Photons" of B. C. Hiesmayr and W. Löffler for a state $\rho_d$ in the "magic simplex" of Bell states \begin{...
3
votes
2answers
740 views

How to evaluate this function?

I’m trying to evaluate y[t] = InverseFunction[ NIntegrate[Sqrt[2]/( -8+Exp[y]+2Exp[-y]), {y,0,t}]] Some of the problems I’m facing, is that i’d like to get the ...
0
votes
1answer
80 views

Substitution into integral [duplicate]

I have to do a substitution (u = pi/2-x) into an definite integral, which I have defined as f[u]. When I evaluate my code, no ...
1
vote
1answer
49 views

Replace second derivative with product rule

Suppose that after some calculations, I find a[x]*D[g[x],x,x]+b[x]*D[h[x],x,x]+c[x]*D[j[x],x] What sort of operations do I need to do in Mathematica such that ...
0
votes
0answers
92 views

Bug in Integrate with Assumptions?

I encountered behaviour I wasn't expecting with Integrate and thought I'd ask here before reporting a bug in case I'm just missing something obvious. A simple example is the following: Computing <...
3
votes
3answers
95 views

Evaluating the error function by integrating $ \mathrm e^{−t^2} $ with Simpson's rule

I am trying to evaluate an error function with Simpson's rule because there is no other way to integrate it. The function is $$ {\rm erf}(x) = \frac{2}{\sqrt π}\int_0^x \mathrm e^{−t^2}\, \mathrm dt ...
2
votes
3answers
118 views

Integration Help

So I have to integrate $$\frac{\sin^n x}{\sin^n x + \cos^n x}$$ and am coding this in Mathematica with (((Sin^n)[x])/(((Sin^n)[x]) + ((Cos^n)[x]))) with the ...
2
votes
2answers
136 views

Poor/Incorrect handling of derivatives of DiracDelta functions in Mathematica

Bug introduced in 12.0 or earlier Integration of $\int_{-\infty}^{\infty} e^{-2|x|}\delta''(x) dx = 2 \int_{-\infty}^{\infty} e^{-2|x|}\frac{\delta(x)}{x^2} dx \rightarrow$ Divergent! Now in ...
0
votes
0answers
43 views

Mathematica does not compute derivative of composite function

I was checking some stuff from my Mechanics Course (non-inertial systems) using Mathematica and I realized I can't get it to compute the second derivative with respect to time. If you could tell me ...
-1
votes
1answer
87 views

Plot fractional trigonometric functions with the Mittag-Leffler function

Can anyone help please? Im trying to plot the solution $X$ of the system as in the paper attached - about fractional calculus which is $X= [E_{\nu}(2t^{\nu})][2 \cos_{\nu}(3t^{\nu})+4 \sin_{\nu}(3t^{...
3
votes
2answers
106 views

Relations between two reciprocal partial derivatives?

My question is similar to How to get the partial derivative of the inverse functions? But they are different. If we have a function $z=z(x,y)$, we can calculate the partial derivative $\left.\frac{\...
3
votes
3answers
319 views

Finding the area between two curves with Integrate

I'm trying to solve and approximate the area between the two graphs. Right now, my functions are stored as f[x_] := 3 Sin[x] g[x_] := x - 1 and then I tried to ...
0
votes
1answer
78 views

Plotting with the Mittag-Leffler function [closed]

I'm trying to plot the solution of fractional differential equations as shown in the photos below, The solutions are in terms of the Mittage-Leffler function, so I evaluated ...
3
votes
0answers
108 views

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

Saw this post and noted that it was posted back in 2013. Just tried V11.3 and got no analytic solution. Just wondering if there is a way for Mathematica to give the desired result? Machine VS Human ...
1
vote
1answer
81 views

Definite Integration by parts

how do I code Integrate[u[x] D[v[x], x], {x, a, b}] to return the following straightforward undergraduate result? ...
1
vote
1answer
73 views

Converting .STL from Shell to Solid for stress analysis of wing

I'm new to Mathematica as this is the first class I've had to used it in my undergrad time as an aerospace engineer. Anyway, I have run into a problem that I can't figure out. I have created .STL's of ...
1
vote
3answers
76 views

How to take a derivative of a function with a real part taken

Consider f[x_]=Re[x^3] I now take the derivative of this D[f[x],x] and the result is ...
1
vote
2answers
45 views

Series expansion of integral

I'm looking for a way to do a series expansion of $$\frac{\mu_0bI_0}{4\pi}\int_0^{2\pi}\frac{\cos\left[\omega\left(t-\frac{1}{c}\sqrt{r^2+b^2-2rb\sin(\theta)\cos(\phi)}\right)\right]}{\sqrt{r^2+b^2-...
2
votes
1answer
54 views

How to make a function depend on the solution of an integration?

I have two functions which are the linear and non-linear solution of an equation. I'm trying to calculate where does the linear solution is a good solution of my equation, when compared with the non-...