# 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.

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### How to mark non-pre-defined functions as functions of a certain variable during calculation of derivative?

If I have some calculation $X=v\cdot\delta$ and I want to calculate $\frac{\partial}{\partial v}X$, the result according to Mathematica is $\delta$. Now, I want to define $\delta$ as a function of $v$ ...
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### Why do I get the error message as integral does not converge?

Can anyone please help me to find the integral value of the following expression, Integrate[(x - t)^(-1), {t, 0, x}] It will be of a big help to me.
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### Evaluating an integral combining a Bessel function with some other functions [closed]

How can I evaluate the integral of $j_1 ^2(x)\exp(-bx)/x$ from 0 to ∞?
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### Variable dimensional integrals

I need to Integrate a function, but let the number of dimensions vary. Integrate[Product[Exp[-( t[[i]] )^2], {i,1,4}], options] ...
104 views

### Evaluating FourierTransform like integral manually

I am trying to evaluate Integrate[x^2*Exp[I k (x - 1)], {k, -∞, ∞}, {x, -∞, ∞}] Since $\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ik(x-1)} d{k}$ is $\delta(x-1)$, ...
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### Limit of an integral and hypergeometric function

I want to evaluate the following integral: Integrate[Sin[θ]^(D1 - Nc - 1)/(A Cos[θ] - I ϵ)^(N1 - Nc), {θ, 0, π}, Assumptions -> A > 0 && ϵ > 0] ...
141 views

### Compute unstable integral with high precision

I want to calculate $$\int_{\mathbb{R}^n} e^{-t_1^4-t_2^4- ...-t_n^4-(1-t_1-t_2-...-t_n)^4} \text{d}t_1 \text{d}t_2 ... \text{d}t_n$$ with the highest possible n. I tried with ...
443 views

### How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$?

How can I simplify $\lim_{n\to \infty } \, \int_0^n e^{-x} \left(1-\frac{x}{n}\right)^n \, dx$ ? Limit[Integrate[(1 - x/n)^n/E^x, {x, 0, n}], n -> Infinity] ...
494 views

### How do I find and plot the intersection of these three surfaces?

I know how to find the region of intersection of these cylinders using integration, but how would I plot that in mathematica? This is what I have ...
82 views

### Problem in Integration

I want to calculate both of these integrals: Integrate[1/Sqrt[c x + Sqrt[ b x + x^2] + a], x] and ...
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### Calculation of sum \begin{aligned}\sum_{k = 1}^{n - 1}\end{aligned}\left(1+\cos\left(\frac{k\,\pi}{n}\right)\right)^n

Having established that Mathematica cannot calculate the following summation: sum = Sum[(1 + Cos[k Pi/n])^n, {k, 1, n - 1}] I implemented the classic "plan B", ...
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### When to trust Integrate?

I'm trying to get an analytical expression for the following integral: ...
167 views

### On Certain Characteristic Function

I am trying to find the characteristic function for Johnson's SU distribution by integrating the probability density function with Exp[I*t*x] but Mathematica is ...
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### Is there any way to do this integral without expanding the exponential power?

Integrate[ x^(1/2) Exp[-10.7* c/x] Sqrt[(1 - (125/1000)^2) (1 - (2*4.18/125)^2) - (1 - (2 x/ 1000))^2], {x, 5.03, 994.97}] or can i do this indefinite ...
108 views

### Problem with minimization of $x^z$

I have plugged in the following problem Minimize[{x^z, z >= 1, z <= 10, x >= 1}, {x, z}] Can anyone explain please why the software does not provide the ...
38 views

### How to deal with the derivative of an variable upper limit integration by the symbolic form?

I need a rusult with the following form D[Integrate[f[t], {t, low[x], upper[x]}], x] (*Out[]=-f[low[x]] (low^\[Prime])[x]+f[upper[x]] (upper^\[Prime])[x]*) My ...
36 views

### Interesting output in Integrate [closed]

I found interesting behaviour, best illustrated in the following example. When trying to evaluate Integrate[1/(d^2 - 1), d] Mathematica 11.3 gives the following ...
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### Fitting a Curie-Weiss law function with a diverging point to a data

I would like to fit a function to some experimental data such that the extracted value of Curie temperature (Tc) describes the temperature at which the response flattens out to a constant value. ...
23 views

### Problem with integrating special function under assumption

i found different results when integrating a special function (see below), depending where i place my assumption (x > 0, x0 > 0). The problem is that the two solutions are not compatible. In fact, if ...
64 views

### Integrate a function of NDSolve

I'm having this problem of integrating a function that's the solution of a NDSolve multiplied by another function. I basically compute a cycle wich add at each step the quantity ...
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### Limit function and DirectedInfinity as output: Interpretation

I tried to compute the following limit: In:= Limit[-A/2 x - 2 x, x -> +Infinity] Out= DirectedInfinity[-2 - A/2] Does the output tell me that the ...
84 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 ...
35 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 ...
65 views

### Very different results of NIntegrate using different methods: which one to believe

Consider the following function: ...
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 ...