calculus – Tricky complex definite integral

I have a complex integral $I$ to solve to show that

$I=int_{-infty}^{infty}frac{x^3sin{x}dx}{x^4+5x^2+4} = frac{(4-e)pi}{3e^2}$

I know that you can integrate over the contour of a semicircle where and represent sin as an infinite sum but not sure what to do next. Should the polynomial be converted to a partial fraction too?

Double integral with complex functions

Suppose $f(z_1,z_2)$ is locally integrable on $mathbb{C}^2$. Is it true that
$$frac{1}{2pi}int_0^{2pi}left(frac{1}{2pi}int_0^{2pi}f(e^{itheta}, e^{it})dthetaright)dtleq frac{1}{2pi}int_0^{2pi}f(e^{itheta}, e^{itheta})dtheta?$$

integration – Splitting a single integral

After performing a dot product inside the integral this result was obtained. What rule was used to split this integral into two integrals in the last step?

$$int_a^{b} textbf{A} cdot textbf{dl} = int_a^{b} (xy dx + (3x-y^2) dy) = int_a^{b} xy dx + int_a^{b} 3x-y^2 dy $$

calculus and analysis – Parametric Surface Integral Boundary Code?

Use the definition of surface integral.

x(u_, v_) = u Cos(v);
y(u_, v_) = u Sin(v);
z(u_, v_) = v;
r(u_, v_) = {x(u, v), y(u, v), z(u, v)};
Integrate(
 y(u, v)*Norm@Cross(D(r(u, v), u), D(r(u, v), v)), {u, 0, 1}, {v, 
  0, π})
(* Integrate(
 y(u, v) Sqrt(# . #) &@Cross(D(r(u, v), u), D(r(u, v), v)), {u, 0, 
  1}, {v, 0, π})*)

2/3 (-1 + 2 Sqrt(2))

Reply the second question.

I believe that $y$ should be $u^2-v^2$ since if $y=u^3-v^2$ then the region is not a simple surface and the integral became so complicated.

x(u_, v_) = 2 u*v;
y(u_, v_) = u^2 - v^2;
z(u_, v_) = u^2 + v^2;
r(u_, v_) = {x(u, v), y(u, v), z(u, v)};
Integrate((x(u, v) + y(u, v) + z(u, v)) Sqrt(# . #) &@
  Cross(D(r(u, v), u), D(r(u, v), v)), {u, v} ∈ Disk())

(4 Sqrt(2) π)/3

x(u_, v_) = 2 u v;
y(u_, v_) = u^2 - v^2;
z(u_, v_) = u^2 + v^2;
r(u_, v_) = {x(u, v), y(u, v), z(u, v)};
reg = ParametricRegion({r(u, v), u^2 + v^2 <= 1}, {u, v});
Integrate(x + y + z, {x, y, z} ∈ reg)

(2 Sqrt(2) π)/3

Clear(x, y, z, u, v, r, θ);
x = 2 u*v /. {u -> r*Cos(θ), v -> r*Sin(θ)};
y = u^2 - v^2 /. {u -> r*Cos(θ), v -> r*Sin(θ)};
z = u^2 + v^2 /. {u -> r*Cos(θ), v -> r*Sin(θ)};
Integrate((x + y + z)*Sqrt(# . #) &@
  Cross(D({x, y, z}, r), D({x, y, z}, θ)), {r, 0, 
  1}, {θ, 0, 2 π})

(4 Sqrt(2) π)/3

x = 2 u*v;
y = u^2 - v^2;
z = u^2 + v^2;
z^2 == x^2 + y^2 // Simplify
(* True *)
Clear(x, y, z, reg)
reg = ImplicitRegion(
   z == Sqrt(x^2 + y^2 ) && 0 <= z <= 1, {x, y, z});
Integrate(x + y + z, {x, y, z} ∈ reg)

(2 Sqrt(2) π)/3

How to solve the attached integral analytically?

How to solve this integral analytically as well as numericallyIntegral Solving

Evaluating definite integral of $(1-x^(18))^(1/20)$ – $(1-x^(20))^(1/18)$

View post on imgur.com

Sorry for bad quality of image, it is power to 18 and 20,

I’m clueless, how to solve this?

calculus – Calculate space using an integral

For example we have the functions:
$y=1/3(2-x)$
$y=6-3x$
$y=x+2$
The intersections points of the functions are: $x=2,1,-1$
But when I want to calculate the integral from -1 to 1 and from 1 to 2.
How can I know which functions belong to which integral? Should I look at the functions that are cut’s in -1 or in 1 (for example in the first integral)?

real analysis – How to prove that this integral exists as both improper riemann integral and as lebesgue integral

$int_{0}^{infty} x^p e^{-x^q} dx $ (p>0 ,q>0),

For riemann integral I $int_{0}^{1} x^p e^{-x^q} dx $+$int_{1}^{infty} x^p e^{-x^q} dx $.

For the 2nd integral I thought of using Comparison testt with 1/x^2 but I will get the limit 0 if q>1 so comparison test with this can’t be used.

So, I am confused with which exptression I should compare and need help.

Also, For LEbesgue integral I need to prove that $int_{0}^{b}|x^p e^{-x^q} |dx<M $for all b there exists an M >0 . For this part I used integration by parts but I am getting this: $(x^p e^{-xq} times 1/(-q) times x^{1-q})_{0}^b + p/q int_{0}^{b} x^{p-q} e^{-xq} dx$. the integral in this expression will be recursive and will appear even after n times performing of integration.

This was my attempt.
So, I need help in both of the qwiuestions.

recursion depth of 1024 error in solving a complicated integral

I want to solve this problem where I need to compute an unusual integral. I wrote this code but there is an error about "recursion depth of 1024 exceeded…".
This is my code:

could you please help me how to solve this problem?
enter image description here

calculus and analysis – Help on script for parametric surface integral

I’m working on a short script that I can repeatedly use to solve parametric surface integrals. There is a post on this elsewhere on MSE, but it is a complicated doublecontourintegral function. It is impossible to parse for a newcomer like me, and it seems possible to create something more simple.

Here’s an example problem:

Find the surface integral of $(x + y + s)dS$, where S is the parallelogram with parametric equations $x = u + v$; $y = u – v$; $z = 1 + 2u + v$; and $0 leq u leq 2$, $0 leq v leq 1$.

So I wanted to write a script that I can just type in the non-parametrized equation, i.e., $(x + y + s)$ in this example. Then I’d enter in the parameters for x, y, and z (given in this problem).

Next, I’d take the partial derivatives with respect to u an v of the parametrized equation.

Next, I’d cross product those two partial derivatives and find its magnitude, using Norm.

Next, I’d re-parametrize $(x + y + s)$.

Finally, I’d integrate the re-paremetrized surface, multiplied by the magnitude of the partials, over the region, given in this problem as $0 leq u leq 2$, $0 leq v leq 1$.

I am new to Mathematica, and I started this sequence with the following code, but it doesn’t seem to work. I’d welcome any help.

Clear(x, y, u, v)
x(u_, v_) := u + v;
y(u_, v_ ) := u - v;
z(u_, v_) := 1 + 2 u + v;
R(u_, v_) := {x(u_, v_), y(u_, v_), z(u_, v_)}

UPDATE: This code works well on problems I’ve tried. But I’m having trouble with another problem. The book answer is 1/(10sqrt(2)). I’m attaching as a picture its worked example. But both of your methods give 1/5sqrt(2). Did I miss something? Is the book in error? Here’s the code I used, and the book’s answer.

x(u_, v_) := u Cos(v);
y(u_, v_) := u Sin(v);
z(u_, v_) := u;
r(u_, v_) := {x(u, v), y(u, v), z(u, v)};
reg = ParametricRegion(r(u, v), {{u, 0, 1}, {v, 0, Pi/2}});
Integrate(x y z, {x, y, z} (Element) reg)

And here’s is the book answer, which is Problem 6:

enter image description here