# Evaluating an integral symbolically

I am trying to integrate following integral symbolically via integrate command:

$$(0.09)\Bigg[1 + \Bigg\{ \int_{3t-4k-7}^{2k+7/3} (0.09) \text{exp}\Bigg(\int_{3s-4k-7}^{2k+7/3} (0.09) \text{exp}\Bigg(\int_{3u-4k-7}^{2k+7/3}(0.09)d\xi \Bigg)du\Bigg)ds\Bigg\} \Bigg]$$

but it gives a result like this:

$$0.09(1+0.06(-1.56+1.23\text{ExpIntegralEi}[1482e^{4.82k-2.43t}]$$

and I can't evaluate this expression further even if I substitute integers with $$k$$.

How can I proceed further of this expression?

Edit: Mathematica code

integral1 = \!$$\*SubsuperscriptBox[\(\[Integral]$$, $$3 u - 4 k - 7$$, $$2 k + 7/3$$]$$\((0.09)$$ \[DifferentialD]\[Xi]\)\)

integral2 = \!$$\*SubsuperscriptBox[\(\[Integral]$$, $$3 s - 4 k - 7$$, $$2 k + 7/3$$]$$\((0.09)$$ Exp[integral1] \[DifferentialD]u\)\)

integral3 = \!$$\*SubsuperscriptBox[\(\[Integral]$$, $$3 t - 4 k - 7$$, $$2 k + 7/3$$]$$\((0.09)$$ Exp[integral2] \[DifferentialD]s\)\)
$$$$

• Please paste copy&pastable Mathematica code instead of LateX. It is not fair to the people who are trying to assist to make them need to retype all your content based on some LateX code.? – Mariusz Iwaniuk Jul 14 '19 at 11:00
• @MariuszIwaniuk Code added. – mrtkp9993 Jul 14 '19 at 11:08

## 1 Answer

Try:

$Version (* "12.0.0 for Microsoft Windows (64-bit) (April 6, 2019)" *) Integrate[(9/100)*Exp[Integrate[(9/100)*Exp[Integrate[9/100, {ξ, 3*u - 4*k - 7, 2*k + 7/3}]], {u, 3*s - 4*k - 7, 2*k + 7/3}]], {s, 3*t - 4*k - 7, 2*k + 7/3}] (*ConditionalExpression[-(1/9) E^(-(E^(21/100)/3)) (ExpIntegralEi[E^(21/25)/3] - ExpIntegralEi[1/3 E^(3/100 (280 + 162 k - 81 t))]), E^(162 k - 81 t) >= 0]*) %[] /. k -> 1 /. t -> 1(*I assume for k and t *) (* -(1/9) E^(-(E^(21/100)/3)) (ExpIntegralEi[E^(21/25)/3] - ExpIntegralEi[E^(1083/100)/3]) *)  $$-\frac{1}{9} e^{-\frac{e^{21/100}}{3}} \left(\text{Ei}\left(\frac{e^{21/25}}{3}\right)-\text{Ei}\left(\frac{e^{1083/100}}{3}\right)\right)$$ • But I want to evaluate it symbolically, not numerically. Is this possible? – mrtkp9993 Jul 14 '19 at 11:29 • @mrtkp9993. I edited the answer. – Mariusz Iwaniuk Jul 14 '19 at 11:50 • My main problem is, I can't get rid of ExpIntegralEi function. – mrtkp9993 Jul 14 '19 at 11:53 • @mrtkp9993.You can't get rid of ExpIntegralEi function ,because is a special function like Erf,BesselJ and so on.It's impossible.You can't simplifying more,try:ExpIntegralEi // FullSimplify`,this is the final answer. – Mariusz Iwaniuk Jul 14 '19 at 12:09 • @mrtkp9993 This is like evaluating$\int_1^x\frac{1}{x}=\log x$, and asking how to remove the$\log\$; you just can't. – AccidentalFourierTransform Jul 14 '19 at 19:24