Calculation and analysis – step-by-step solution not available ((possible intermediate steps))

When using the (( WolframAlpha("integrate 1/(2+Sqrt(1-x)+Sqrt(1+x))"))) Function to find integration steps, this message appears to me, what is the explanation and why?

As an example of $ int frac {1} { sqrt {1-x} + sqrt {x + 1} +2} dx $

$$
int frac {1} {2+ sqrt {1-x} + sqrt {1 + x}} , dx approx left ( sqrt {x + 1} + left (- frac {1 )} { sqrt {x + 1} +1} -1 right) sqrt {1-x} + frac {1} { sqrt {x + 1} +1} – frac {2 left ( 0.707107 sqrt {x + 1} right)} { sin} right) + C.
$$

I don't understand the steps

Enter the image description here

Why does that happen?

Objective – calculation of the small projection distance

I'm trying to build a small slide projector, but since I'm not a lens, I can't get a properly focused image.

I am trying to achieve this:
Enter the image description here

I have a 20 mm lens (biconvex?) With a focal length of 50.8 mm. So when I calculate the distance, I need a distance of about 81.28 mm, but whatever I tweek doesn't focus.

Apparently I'm doing something wrong here or do I need a different lens type?

Thank you in advance!

Calculation and analysis – limit at the infinity of arbitrary functions

Here I have the code that takes the limit of an expression

Limit((-I E^(I x) f1(y))/(g2^(Prime)(Prime))(y), x -> (Infinity) )

and the output returned is INDETERMINATE while the desired output is $ infty $. Or if I did that instead

Limit((g2^(Prime)(Prime))(y)/(-I E^(I x) f1(y)), x -> (Infinity) )

I would want to get 0 and not INDETERMINATE.

How would I let Mathematica know that $ f $ and $ g $ Functions are irrelevant when evaluating the limit?

Thanks for any help.

Calculation and analysis – how do you get the step-by-step derivation from it?

I am a newcomer to Mathematica and I wanted to use it to solve some questions for which I do not have the mathematical support to solve them on paper. I want to take this equation and have it as a partial derivative. However, I also want a step-by-step guide to help me hold out so I know what's going on. Instead, I get this error message. Could someone please help?

WolframAlpha("D(-T0/ ProductLog(-E^(-1+dS/dC), dS)")

WolframAlpha::timeout: The call to WolframAlpha(D(-T0/ ProductLog(-E^(-1+dS/dC), dS)) has exceeded 30. seconds. Increasing the value of the TimeConstraint option may improve the result.

Thank you so much!

Calculation and analysis – numerical solution to the triple integration of a region with variable borders?

I am asked to calculate the mass of the region bounded by the plane above (2x + 3y - z = 2) and below by the triangle on the xy plane with corner points (0,2),(1,0),(4,0). The density is proportional to the distance from the plane to the xy plane, so that's the function d(x,y,z) = kz.

My limits are:

0 ≤ z ≤ (2x + 3y - 2)

(1 - 0.5y) ≤ x ≤ (4 - 2y)

and 0 ≤ y ≤ 2.

My code is:

Integrate(d(x,y,z),{z,0,2x+3y-2},{x,1-0.5y,4-2y},{y,0,2})

But Mathematica doesn't return a number (if I'm correct it should be 19k).

Instead I get: k(3-1.5y)(-2+2x+3y)^2.

However, if I integrate the function step by step:

Integrate(Integrate(Integrate(d(x,y,z),{z,0,2x+3y-2}),{x,1-0.5y,4-2y}),{y,0,2}),

I get a numerical value.

Is there a specific syntax that Mathematica needs that I don't know?

Calculation of the maximum module

For each $ a in mathbb C $ define $ f_a: mathbb C to mathbb C $ by $ f_a (z) = exp (z) -a $. I am mainly interested in real values $ a in (- infty, -1) $.

For each $ r in (0, infty) $ define $ M_a (r) = max {| f_a (z) |: | z | = r } $. In order to $ M_a (r) = | z & # 39; | $ for a point $ z & # 39; $ in the picture of the circle with the radius r. Is there a more explicit (numerical) formula for $ M_a (r) $ in terms of $ a $ and $ r $?

Calculation and analysis – How do I deal with unit integration?

I have defined some functions that units should use. Now another function is used to integrate it, whereby the integration variable is also quantified and the input into this function takes place. I better just show a code:
Function definition

It is not beautiful, but I think that nothing is beautiful in Mathematica. I don't speak the tungsten language very well and I don't have much experience with symbolic expressions.

Back to the code: this will work fine, but if I plug in some numbers or variables, it won't work:
The first two functions without integration work perfectly
The first integration function triggers errors
The second integration raises errors

As you can see, something is wrong. I did several hours of research yesterday and just can't get it to work. I want to draw this and do some calculations, so nothing special, but for this to work, it has to accept inputs like 1 s and t s.

Bonus: I can't take care of these delayed features. As you can see, h (x, s, t) gives a simple piecewise function. So why does it take ages and millennia to plan this? I have to do h(x_, s_, t_) = ... instead of :=? That didn't seem to do anything for me. This also assumes that I have not used any units so the above works.

The code in the text:

Clear("Global`*")
g = UnitConvert(
   Entity("Planet", "Earth")(EntityProperty("Planet", "Gravity")));
TWR(t : Quantity(_, unit_)?(CompatibleUnitQ(#, "Seconds") &)) := 
 FullSimplify(
  Piecewise({{0.65 Quantity(1, (1/("Seconds")))*t, 
     0 Quantity(1, "Seconds") < t < 
      2.05 Quantity(1, "Seconds")}, {1.3325, 
     t >= 2.05 Quantity(1, "Seconds")}}), t (Element) Reals)
a(t : Quantity(_, unit_)?(CompatibleUnitQ(#, "Seconds") &)) := 
 FullSimplify((TWR(t) - 1) g, t (Element) Reals)
v(t : Quantity(_, unit_)?(CompatibleUnitQ(#, "Seconds") &)) := 
 FullSimplify(
  Integrate(
   a(Quantity(T, "Seconds")), {Quantity(T, "Seconds"), 
    Quantity(0, "Seconds"), t}, Assumptions -> t (Element) Reals), 
  t (Element) Reals)
h(x_, s_, t : Quantity(_, unit_)?(CompatibleUnitQ(#, "Seconds") &)) :=
  FullSimplify(
  x Quantity(1, "Meters") + 
   s Quantity(1, (("Meters")/("Seconds")))*t + 
   Integrate(
    v(Quantity(c, "Seconds")), {Quantity(c, "Seconds"), 
     Quantity(0, "Seconds"), t}, 
    Assumptions -> t (Element) Reals && c (Element) Reals), 
  t (Element) Reals)

Calculation and analysis – solving multiple integrals with symbolic boundaries

I try to solve a double integral, although I'm not sure how I could possibly deal with the symbolic boundaries.
I tried the following code:

i2 = (xprime^2*yprime^2)/2
xminp = x - 1/2 (y - yprime);
xmaxp = x + 1/2 (y - yprime);
yminp = 0 ;
ymaxp = y ;
I2 = NIntegrate(i2, {xprime, xminp, xmaxp}, {yprime, yminp, ymaxp})

However, this leads to the following error:

NIntegrate: xprime = x +1/2(-y +yprime) is not a valid limit of integration)

Does anyone know how I can do this symbolic integration in Mathematica?
Thanks a lot.

Database – Draft calculation chart with tables

In general, ER diagrams can be used when designing the dependency between tables, and there are various tools for drawing them. For the chain of calculations with simple variables, there is the concept of the calculation graph.

Is there an industry-standard design technique or a tool that can be used to visualize calculations with different columns in different tables? The operations include standard table operations: lookup, pivot, totals, etc. Or is the ER diagram the best way to deal with it?