# 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 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 ...
2answers
921 views

### Integral with the derivative of Dirac delta does not evaluate correctly?

Integrate[DiracDelta'[y - z] DiracDelta[x - z], {z, -Infinity, Infinity}, Assumptions -> x \[Element] Reals && x \[Element] Reals]
2answers
229 views

### Differentiate the product of some terms

How can I compute the following derivative, $$\frac{\partial}{\partial \lambda_j} \prod_{i=1}^k (1+\lambda_i)e^{\lambda_i} \quad \text{for }\; 1\le j \le k$$ for some positive integer $k$ which is ...
1answer
434 views

### Asymptotic Rate of Growth

How can I calculate the Asymptotic Rate of growth of a function, for instance like: $X^3 - X^2 - X -1$ EDIT: For instance, as you can see in this graph, after the 1200 the function approximates to ...
2answers
1k views

### Sum of series with log in each term [closed]

I was solving recurrence relation of Introduction to Algorithms by CLRS, 3rd. edition. Problem 4-3 (i) $$T(n) = T(n-2) + \frac{1}{lg \; n}$$ I tried few ways, like expending with iteration method. ...
1answer
1k views

### How to define a derivative/Variation operation in Mathematica from scratch

I am looking for a method in Mathematica 8 to define my own derivative/variation (independent of built-in operations). I do not want to re-define built-in objects since I fear that this could cause ...
1answer
572 views

### How to solve this integral

I'm trying to deal with this article, but I don't understand how equations 5.9 and 5.10 were obtained. I tried to use Mathematica 8 to solve it, but the answer was different. I have a problem with ...
2answers
8k views

### How to find (numerical) value of a derivative at point?

I have the following function: f[0, 0] = 0 f[x_, y_] := Exp[-(x^2 + y^2)^(-1)] How do I find its partial derivatives at any given point, including $(0,0)$? This ...
1answer
459 views

### ReplaceAll[] and Limit[] don't give correct results for this expression under extreme variables [duplicate]

Possible Duplicate: Funny behaviour when plotting a polynomial of high degree and large coefficients 1/x^2 + (3 + x)/(6 (1 - Exp[x] + x)) ——This is a ...
5answers
2k views

### How to make a Line[] with no end?

I'm trying to do this: In this graph, the secant points are aproximated in order to become the tangent, it seems I need some kind of function which plots a line based on two points and it's points ...
1answer
479 views

### How to move differentation before the integral?

I have quite a lot expressions that need to be integrated after switching the integral sign with differentiation operator. The additional problem is that the bounds are dependent to one of the ...
3answers
361 views

### Need help evaluating definite integral to a function of Y

Suppose $Y = \sqrt{2T}\cos(U)$, $0 \le u \le \pi$, and $0 \le \cos^{-1}(\frac{y}{\sqrt {2t}}) \le \pi )$, so $-1 \le \frac{y}{\sqrt{2t}} \le 1$, with all $\mathbb{R}$. The iterated integral ...
3answers
611 views

1answer
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### How to collect q[t] from the following integration

As shown in the following program, the q[t] in a can be collected from the integration by defining the integration of ...
1answer
130 views

### How to explain the results of the following integrals?

Why does the following program lead to different results of a21 and a211? ...
0answers
422 views

### Simple contour integral with a parameter gone wrong

Bug introduced in 7.0 and fixed in 7.0.1 I run into the following problem, I tried to evaluate a very simple integral: ...
2answers
303 views

### How to simplify the integration result of the following program?

In the following program, the fec[x] is a function depending on $x$ while q3[t] and q5[t] ...
3answers
13k views

### How to find the nth derivative?

This question is not the same as my last one. How do you find the $n$-th derivative where $n$ is a variable? For example, you can find the nth derivative for a specific $n = 3$ ...
2answers
585 views

### How to iteratively display the nth derivative?

I am new to Mathematica, but not so new to programming. What I'd like to do in Mathematica (not using an API) is to display on each line, the nth derivative of a function, e.g. for n = 1..10. Is there ...
2answers
3k views

2answers
5k views

### How to convert a system of parametric equations to a normal equation?

For example, I have a system of parametric equations (R is a constant number) : ...
1answer
347 views

### Extra factors appear when evaluating Euler integrals

Note: this is fixed in version 9. When I perform the double integral in Mathematica, Integrate[(x (1 - x))^z (y (1 - y))^z, {x, 0, 1}, {y, 0, 1}] which should ...
6answers
5k views

### Generating evenly spaced points on a curve

In the KnotData package a simple command such as points = Table[KnotData[{3, 1}, "SpaceCurve"][t], {t, 0, 2 Pi, 0.1}]; will ...
2answers
1k views