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I have an integral that depends on two parameters ##a\pm\delta a## and ##b\pm \delta b##. I am doing this integral numerically and no python function can calculate the integral with uncertainties.

So I have calculated the integral for each min, max values of a and b.

As a result I have obtained 4 values, such that;

$$(a + \delta a, b + \delta b) = 13827.450210 \pm 0.000015~~(1)$$

$$(a + \delta a, b - \delta b) = 13827.354688 \pm 0.000015~~(2)$$

$$(a - \delta a, b + \delta b) = 13912.521548 \pm 0.000010~~(3)$$

$$(a - \delta a, b - \delta b) = 13912.425467 \pm 0.000010~~(4)$$

So it is clear that ##(2)## gives the min and ##(3)## gives the max. Let us show the result of the integral as ##c \pm \delta c##. So my problem is what is ##c## and ##\delta c## here?

The integral is something like this

$$I(a,b,x) =C\int_0^b \frac{dx}{\sqrt{a(1+x)^3 + \eta(1+x)^4 + (\gamma^2 - a - \eta)}}$$

where ##\eta## and ##\gamma## are constant.

Note: You guys can also generalize it by taking ##\eta \pm \delta \eta## but it is not necessary for now.

I have to take derivatives or integrals numerically. There's no known analytical solution for the integral.

##\eta = 4.177 \times 10^{-5}##, ##a = 0.1430 \pm 0.0011##, ##b = 1089.92 \pm 0.25##, ##\gamma = 0.6736 \pm 0.0054##, ##C = 2997.92458##

So I have calculated the integral for each min, max values of a and b.

As a result I have obtained 4 values, such that;

$$(a + \delta a, b + \delta b) = 13827.450210 \pm 0.000015~~(1)$$

$$(a + \delta a, b - \delta b) = 13827.354688 \pm 0.000015~~(2)$$

$$(a - \delta a, b + \delta b) = 13912.521548 \pm 0.000010~~(3)$$

$$(a - \delta a, b - \delta b) = 13912.425467 \pm 0.000010~~(4)$$

So it is clear that ##(2)## gives the min and ##(3)## gives the max. Let us show the result of the integral as ##c \pm \delta c##. So my problem is what is ##c## and ##\delta c## here?

The integral is something like this

$$I(a,b,x) =C\int_0^b \frac{dx}{\sqrt{a(1+x)^3 + \eta(1+x)^4 + (\gamma^2 - a - \eta)}}$$

where ##\eta## and ##\gamma## are constant.

Note: You guys can also generalize it by taking ##\eta \pm \delta \eta## but it is not necessary for now.

I have to take derivatives or integrals numerically. There's no known analytical solution for the integral.

##\eta = 4.177 \times 10^{-5}##, ##a = 0.1430 \pm 0.0011##, ##b = 1089.92 \pm 0.25##, ##\gamma = 0.6736 \pm 0.0054##, ##C = 2997.92458##

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