Please, I am trying to solve definite integral and its work but the results not simplified.
F = Integrate[(x^4*Exp[x])/(Exp[x] - 1)^2, {x, 0, 44}]
This is the result
(1/(15 (-1 + E^44)))4 (-\[Pi]^4 +
E^44 (\[Pi]^4 +
30 (-1874048 + 42592 Log[-1 + E^44] - 2904 PolyLog[2, 1/E^44] -
132 PolyLog[3, 1/E^44] - 3 PolyLog[4, 1/E^44])) +
30 (-42592 Log[-1 + E^44] +
3 (468512 + 968 PolyLog[2, 1/E^44] + 44 PolyLog[3, 1/E^44] +
PolyLog[4, 1/E^44])))
You need to convert the result to a numerical value.
F = Integrate[(x^4*Exp[x])/(Exp[x] - 1)^2, {x, 0, 44}];
N[F]
25.9758
Visualisation
Show[Plot[(x^4*Exp[x])/(Exp[x] - 1)^2, {x, -4, 48}],
Plot[(x^4*Exp[x])/(Exp[x] - 1)^2, {x, 0, 44},
Filling -> Axis, PlotRange -> All]]
Related
enter image description here
I failed to do the following expression and make it give accurate results if any one can help me I will be glade. I attached my expression in a pic "want this" and my trial as "my trial". the correct answer must equal 0.119 when a=1, b=10, m=3, n=6. thanks a lot in advance.
a = 1
b = 10
m = 3
n = 6
a^1 b^n (Sum[
Sum[Sum[Sum[(-1)^(k + v - n + m + 1)
If[k == 0, 1,
SeriesCoefficient[Series[(-Log[1 - x])^k, {x, 0, 30}],
p + k]] If[n - k - 2 == 0, 1,
SeriesCoefficient[
Series[(-Log[1 - x])^(n - k - 2), {x, 0, 30}],
q + (n - k - 2)]]
Binomial[n - m - 1, k] Binomial[b - 1,
v] (-PolyGamma[0, -1 + 1/a - k + n + q] +
PolyGamma[0, 2/a + n + p + q + v])/(a (1 + k + p + v) +
1), {q, 0, 30 - (n - k - 2)}], {p, 0, 30 - k}], {v, 0,
b - 1}], {k, 0, n - m - 1}])/((m - 1)! (n - m - 1)!)
I found the solution for the problem. the problem was when the value of k was 0 the coefficient will not equal 1 but the whole expression must be found from the start for a value of k that will start from 1 and an expression when the value of k is 0. yet I failed to solve it using MATHEMATICA but by doing the above I succeed to get the correct result. thank you all for your precious time and opinions.
I'm trying to plot b against a with the following equation
2 *r* (sin[b]^2) * cos[b] * sin[a + b] == sin[a + 2*b]*((sin[a]^2) + r*(sin[b]^2))
with r at >= 0 and let's say <1000
Solve[2 cos[b] sin[b]^2 sin[a + b] == (sin[a]^2 + sin[b]^2) sin[a + 2 b], b]
However this does give me the error
Solve::nsmet: This system cannot be solved with the methods available to Solve.
How would I be able to solve this equation?
Is this what you are trying to obtain?
Edit
sol = FullSimplify#Solve[
2 r Sin[b]^2 Cos[b] Sin[a + b] == Sin[a + 2 b] (Sin[a]^2 + r Sin[b]^2), b];
Show[Table[Plot[b /. sol[[4]], {a, 3.3, 4.7}],
{r, {0, 0.1, 0.25, 0.5, 1, 2}}],
Frame -> True, Axes -> None,
PlotRange -> {Automatic, {0, Automatic}}]
I'm solving and plotting the equations of motion for the double pendulum using Mathematica's NDSolve.
I've successfully plotted the Angular position using a standard plot. But when I come to use the parametric plot for the position of each mass. I get no errors but simply no plot.
eqn1 = 2 th''[t] + Sin[th[t] - ph[t]] (ph'[t])^2 + Cos[th[t] - ph[t]] (ph''[t]) + (2 g/l) Sin[th[t]]
eqn2 = ph''[t] + Sin[th[t] - ph[t]] (th'[t])^2 + Cos[th[t] - ph[t]] (th''[t]) + (g/l) Sin[th[t]]
eqnA = eqn1 /. {g -> 10, l -> 1}
eqnB = eqn2 /. {g -> 10, l -> 1}
sol = NDSolve[{eqnA == 0, eqnB == 0, th[0] == 0.859, th'[0] == 0, ph[0] == 0.437, ph'[0] == 0}, {th, ph}, {t, 0, 10}]
Plot[{th[t], ph[t]} /. sol, {t, 0, 10}]
r1 = {lSin[th[t]] + lSin[ph[t]], -lCos[th[t]] - lCos[ph[t]]} /. {l -> 1, g -> 10}
ParametricPlot[r1 /. sol, {t, 0, 10}]
Replace
r1 = {lSin[th[t]] + lSin[ph[t]], -lCos[th[t]] - lCos[ph[t]]} /. {l->1, g->10}
with
r1 = {l*Sin[th[t]] + l*Sin[ph[t]], -l*Cos[th[t]] - l*Cos[ph[t]]} /. {l->1, g->10}
and your ParametricPlot should appear.
One useful trick you might remember, when any plot doesn't appear you can try replacing the plot with Table and see what it shows. Often the table of data provides the needed hint about why the plot isn't appearing.
I want to plot a definite integral
myIntegral[x_] := NIntegrate[Sqrt[(a - b)^2 + (c - d)^2]/ (a - b), {a, 0, x}, {b, x, 1}, {c, 0, 1}, {d, 0, 1}]
Plot[myIntegral[x], {x, 0, 1}]
I am using Mathematica 11.3. When I evaluate the code, Mathematica immediately gives some warnings about numerical integration converging too slowly, then it seems stuck in computation.
I have tried some options of NIntegrate, for example as suggested here, but have had no success.
Is there a way to obtain the result of Plot in an acceptable time (e.g. some minutes)?
It is within reach of Integrate.
Assuming[0<x<1,
Simplify[
Integrate[Sqrt[(a-b)^2 + (c-d)^2]/(a-b), {a,0,x}, {b,x,1}, {c,0,1}, {d,0,1}]]]
which gives you
(-11*Sqrt[2] + 11*Sqrt[2+(-2+x)*x] + x*(-8-7*Sqrt[2+(-2+x)*x] + 13*Sqrt[1+x^2] +
2*x*(6-3*Sqrt[2+(-2+x)*x] + x*(-4+2*x + Sqrt[2+(-2+x)*x] - Sqrt[1+x^2]))) +
18*x^2*ArcCoth[Sqrt[2+(-2+x)*x]] + 9*ArcSinh[1] - 3*ArcSinh[1-x] - 3*ArcSinh[x] +
6*Log[1-x] + 9*x*Log[-1+Sqrt[2+(-2+x)*x]] + 3*((-2+x)*Log[1+Sqrt[2+(-2+x)*x]] -
4*x*Log[-(((-1+x)*(1 + Sqrt[1+x^2]))/x)] + x^3*(Log[2+x^2+2*Sqrt[1+x^2]] +
2*Log[(1-x)/(x+x*Sqrt[2-2*x+x^2])])))/36
And that should be much faster to plot than doing thousands on individual NIntegrate
I have a circle with the expression:
x^2+y^2+10x-14y-7=0
I need to find the radius and the coordinates to the center of the circle using Wolfram Mathematica; in Symbolab it tells me to rewrite the expression in the form of the standard eclipse equation.
I know that the answer should be c=(x+5)^2 + (y-7)^2 = 81, so the radius is sqrt(81) = 9 ...
How is this achievable in Mathematica (I'm very new to this...)?
Thanks and best regards (-:
For example,
SolveAlways[{x^2 + y^2 + 10 x - 14 y - 7 == (x - a)^2 + (y - b)^2 - r^2}, {x, y}]
Solve[{
CoefficientList[x^2 + y^2 + 10 x - 14 y - 7, {x, y}]
==
CoefficientList[(x - xc)^2 + (y - yc)^2 - r^2, {x, y} ],r>0}
, {xc, yc, r}]
{{xc -> -5, yc -> 7, r -> 9}}