# My function has an integral in it. How can I calculate and plot it?

I'm new to using mathematica, but I want to do a plot of this equation below:

$$F(\nu) = \frac{4 \pi h \nu^{4}}{c^{2}} \int_{r_{i}}^{r_{o}} \frac{r d r}{\exp \left[\left(\frac{3 \pi}{2}\right)^{1 / 4} \frac{h \nu}{k T_{*}}\left(\frac{r}{R_{*}}\right)^{3 / 4}\right]-1}$$ where $$R_* = 2.5(696.34*10^6)$$, $$r_i = 6*R_*$$. $$r_0 = 7*10^4*R_*$$, $$T=4000$$, $$h$$ is de Planck constant, $$c$$ is speed of light and $$k$$ is the Bolztmann constant. The plot should be from $$\nu_1 = 10^{10}Hz$$ to $$\nu_2 = 10^{16}Hz$$. All values here are with the right units when using SI, so don't bother with it.

My main problem is that I do not know how can I set a function like this inside Mathematica because it is a function of $$\nu$$ and it has an integral in it. I know how to do integrals, but in this case, I have to compute the integral to each one of the $$\nu$$ values of my domain. I have no clue where to begin. Can someone help me?

Oh, it is supposed to look like this

EDIT: I followed the tip from the first comment but now I have another problem: The mathematica does not compute my integral. As it is indicate in the comment, I did:

fun[a_?NumericQ] :=  NIntegrate[((9.3*(10^(-50))) (a^4)*x)/(e^(6.89*(10^(-24))*a*x^(3/4)) - 1), {x,6*2.5*(696.34*(10^6)), 7*(10^4)*2.5*(696.34*(10^6))}]; Plot[fun[a], {a, 10^10, 10^16}]


but I get a black graph. It got me back a message saying:

The integrand has evaluated to non-numerical values for all sampling points in the region with boundaries

"Further output of NIntegrate::inumr will be suppressed during this calculation"

• fun[a_?NumericQ] := NIntegrate[a*x, {x, 0, 1}] ; Plot[fun[a], {a, 1, 5}]
– I.M.
Commented May 7, 2021 at 2:42
• Thanks! It was a valuable tip!
– kplt
Commented May 7, 2021 at 3:18
• Post the formulas by Mathematica code. Commented May 7, 2021 at 6:36

Clear["Global*"]

R = 25 (69634*^3);
ri = 6 R;
r0 = 7*^4 R;
T = 4000;

h = QuantityMagnitude[Quantity[1, "PlanckConstant"] // UnitConvert];

c = QuantityMagnitude[Quantity[1, "SpeedOfLight"] // UnitConvert];

k = QuantityMagnitude[Quantity[1,
"BoltzmannConstant"] // UnitConvert];

F[v_?NumericQ] := 4 Pi h v^4/c^2 * NIntegrate[
r/(Exp[(3 Pi/2)^(1/4) h v/(k T) (r/R)^(3/4)] - 1),
{r, ri, r0}]

LogLogPlot[F[v], {v, 10^10, 10^16}]


EDIT: Or

Plot[Log10[F[10^v]], {v, 10, 16},
Frame -> True,
FrameLabel -> {"log v  (Hz)", "log F(v)"}]
`