Solve Integral Calculus Problems with Ease

College
Calculus and Analysis
Integral Calculus

I need to evaluate the following integral:
$\int \int_{G} \frac{\ln (x^{2} + y^{2})}{x^{2} + y^{2}} d x d y$

Kyler Oconnor 2022-11-14

How to solve this integration: $\int_{0}^{1} \frac{x^{2012}}{1 + e^{x}} d x$

kaltEvallwsr 2022-11-13

I have to write application which solves task presented below. I only know some c# so I will stick to it. It is some kind of homework but I am asking for help with understanding this and advice for implementation.

Take natural $n >= 1$ and real $a > 1$
differential problem
$y^{'} (x) = f (x, y (x)), y (1) = 0$
where $f (x, y) = \sqrt{y + 1} + x$
solve for $[1, a]$ with step $h = \frac{a - 1}{n}$ with Euler and Heun methods.

Compare results with exact solution $y (x) = x^{2} - 1$ by finding maximum error of each method.

Jadon Johnson 2022-11-13

I need to prove that
$lim_{n \to \infty} \int_{0}^{n^{2}} n \sin (x / n) e^{- x^{2}} d x = 1 / 2$

evitagimm9h 2022-11-13

We are interested in solving the advection equation $u_{t} = u_{x}$ where $0 \leq x < 1$ , $t \geq 0$
with periodic boundary conditions and $u (x, 0) = f (x), f (x) = f (x + 1)$ In the grid $x_{0}, x_{1}, \dots x_{N}$ , $t_{0}, t_{1}, \dots t_{N}$ .
using the forward euler method, or $\frac{v (x_{j}, t_{n + 1}) - v (x_{j}, t_{n})}{Δ t} = \frac{v (x_{j + 1}, t_{n}) - v (x_{j - 1}, t_{n})}{2 Δ x}$ where $v$ is the approximation of $u$ at the grid points.

What I don't understand:
We are essentially using $v (x_{j - 1}, t_{n}), v (x_{j}, t_{n}), v (x_{j + 1}, t_{n})$ to calculate $v (x_{j}, t_{n + 1})$ . Initially this is fine because $v (x_{j}, 0)$ is known for all $j$ , but how would we calculate $v (x_{0}, t_{1})$ ? we can't, since that would require us to know $v (x_{- 1}, t_{0})$ which doesn't exist. And this issue propagates, each new $t_{n}$ we can calculate one less point than $t_{n - 1}$ .
What am I missing?

Audrey Arnold 2022-11-13

Where is this function concave up?
The function is: $(x^{2} - 16)^{6}$ .
So I did all the relevant calculations and all my others answers about inflexion points, critical points, extreme points, concave down, increasing and decreasing intervals were correct. But my interval for concave up keeps coming up as wrong.
This is the interval I calculated:
$(- \infty, - 4) \cup (- 4, - \frac{4 \sqrt{11}}{11}) \cup (\frac{4 \sqrt{11}}{11}, 4) \cup (4, \infty)$
I also tried various alternative ones like:
$(- \infty, - \frac{4 \sqrt{11}}{11}) \cup (\frac{4 \sqrt{11}}{11}, \infty)$

Zackary Diaz 2022-11-13

Use the Euler Method to write down an iterative algorithm so that $y_{n + 1} = y (t_{n + 1})$ can be determined from $y_{n} = y (t_{n})$ , where $t_{n} = n △ t$ is the size of the time step, for the following Ordinary Differential Equation $\frac{d y}{d t} = y^{2} + \sin (t)$

I'm confused about this question because the question's I'm been doing upto this one is that I am given an initial $(x_{0}, y_{0})$ and then I draw up a table to find a $y$ value for a corresponding $x$ value.. so this question is very confusing as I have never came across a question such as this one.

Siena Erickson 2022-11-12

Since we have a simple conversion method for converting from radians to degrees, $\frac{180}{π}$ or vice versa, could we apply this to Euler's Identity, $e^{i π} = - 1$ and traditionally in radians, to produce an equation that is in degrees; one that may or may not be as simplistic or beautiful as the radian version, but yet is still quite mathematically true?

If not, why? If so, how did you derive it?

Nola Aguilar 2022-11-11

I'm trying to solve this 1st order ODE numerically by bringing it into an explicit form, but I don't think it is valid because of the dependency on x_n in the final expression.
$\frac{d y}{d x} + x = 0 \frac{d y}{d x} = - x$
Alpha is the angle from point with index (n) to point with index (n+1).
$t a n (α) = \frac{d y}{d x} t a n (α_{n}) = - x_{n}$
I call h the step size.
$t a n (α_{n}) = \frac{y_{n} - y_{n - 1}}{x_{n} - x_{n - 1}} t a n (α_{n}) = \frac{y_{n} - y_{n - 1}}{h}$
When I rearrange this I obtain the following form.
$y_{n} = y_{n - 1} + h \cdot t a n (α_{n}) y_{n} = y_{n - 1} + h \cdot (- x_{n}) y_{n} = y_{n - 1} - h \cdot x_{n}$
Is this the final numerical solution of this 1st order ODE?
$y_{n} = y_{n - 1} - h \cdot x_{n}$
EDIT: I've brought it to a better form I think.
$y_{1} = y_{0} - h \cdot x_{1} y_{2} = y_{1} - h \cdot x_{2} = y_{0} - h \cdot x_{1} - h \cdot x_{2} y_{3} = y_{2} - h \cdot x_{3} = y_{0} - h \cdot x_{1} - h \cdot x_{2} - h \cdot x_{3}$
So that the final solution is the following. $y_{n} = y_{0} - h \cdot (x_{1} + . . . + x_{n})$

drogaid1d8 2022-11-11

The Cauchy's Beta Integral is given by
$\int_{- \infty}^{\infty} \frac{d t}{(1 + i t)^{x} (1 - i t)^{y}} = \frac{π 2^{2 - x - y} Γ (x + y - 1)}{Γ (x) Γ (y)}$

mxty42ued 2022-11-11

Asymptotes of a function
I have the function $f (x) = x \cdot \arctan (x)$ And i need to find all asymptotes, I am stuck at $lim_{x \to \infty} x \cdot \arctan (x)$ , please help me.

Ayanna Goodman 2022-11-09

Finding the horizontal asymptotes of f
The question I must answer is:
"Find the horizontal asymptotes of $f (x) = \frac{\sqrt{10 x^{2} + 11}}{12 x + 10}$ "
However, I am unsure how to find the horizontal asymptote or what this means. Can anyone explain this?

charmbraqdy 2022-11-08

Hy. Can someone please explain me how can I resolve $\int_{0}^{\infty} e^{- (s I - A) t} d t$ ? I must have the final result $(s I - A)^{- 1}$ . I think it has the euler form, for beta integral but I don't know how can I get to that result.

Emmanuel Giles 2022-11-08

I need to find the following integral:
$\int \frac{1}{(1 + x) \sqrt{1 + x - x^{2}}} d x .$
I tried using Euler's formula and put $x t + 1 = \sqrt{1 + x - x^{2}}$ and after to do integration in parts but that goes nowhere. Is there anotehr method to solve it?

Fillipus P Kasita2022-11-06

If f : [0, ∞) → R is continuous, the Laplace transform of f is the function F defined by F(s) = Z ∞ 0 f(x)e −sx dx and the domain of F is the set consisting of all numbers for which the integral converges. Find the Laplace transforms of the following functions. a) f(x) = 10 b) f(x) = sin 2x c) f(x) = e x d) f(x) =

Kenna Stanton 2022-11-06

$\frac{d x}{d t} = ω \sqrt{A - x^{2}},$
where $A, ω > 0$ and $x = x_{0}$ at $t = 0$ .
It is to be solved from $t = 0$ to $t = 50.0$ . It has analytical solution
$x (t) = \sqrt{A} \sin (ω t + φ),$
where $\sin (φ) = \frac{x_{0}}{\sqrt{A}}$ if $x_{0} \leq \sqrt{A}$ .

The question I am trying to solve is the following
Rewrite the differential equation you have been given at the start of this document in the correct form for applying the Euler and Euler-Cauchy numerical schemes. Write down an appropriate 1 Euler method recursive scheme to solve this differential equation for the following values of the parameters and initial conditions:
$ω = 3.1$ , $A = 10.0$ , $x_{0} = 2.0$ .

Kayden Mills 2022-11-06

Show that $\int_{0}^{π / 2} \frac{\log^{2} \sin x \log^{2} \cos x}{\cos x \sin x} d x = \frac{1}{4} (2 ζ (5) - ζ (2) ζ (3))$

figoveck38 2022-11-05

How can I prove the monotonicity of a function?
Given is the function $f (x) = \frac{x}{\sqrt{1 - x^{2}}}$ .
How can I prove that this function is monotonic and thus injective?

Madison Costa 2022-11-05

Find the asymptotes of the Parametric equation?
Consider
$x (t) = 2 e^{- t} + 3 e^{2 t}$
$y (t) = 5 e^{- t} + 2 e^{2 t}$
which represents a non rectilinear paths
Horizontal and Verical Asymptotes :
If $t \to + \infty o r - \infty$ and $x (t) a n d y (t) \to \infty$ , So there are no asymptotes parallel to coordinate axis
oblique Asymptotes:
Please tell me how to find the Oblique asymptotes

pin1ta4r3k7b 2022-11-04

$\int e^{- 3 t} c o s (2 - \sqrt{3} t) d t$
I have been asked to evaluate that using complex exponential/euler's method. I have done many similar questions but all of them had something like (cos3x), sin(5t) etc. This is the first time I have come across a question where its of the format (cos a+bx) and cannot understand how to deal with the extra term. I have been stuck for a while now and have thus decided to ask the community for help, please help me out.

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