# 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 ...
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### 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]
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### 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 ...
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### 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 ...
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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. ...
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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...
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 ...
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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 ...
130 views

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

Why does the following program lead to different results of a21 and a211? ...
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: ...
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### 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] ...
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### 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 $n = 3$ ...
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 ...
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### 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) : ...
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 ...