Integrals that cannot be solved with a direct approach

Question

I'm collecting problems for which the basic or direct approach in Mathematica (v. 13 or later) does not work, but requires some trick or mathematical insight. I'd like to start with integration.

For instance,

int = Integrate[
Sqrt[Log[9-x]]/(Sqrt[Log[9-x]]+ Sqrt[Log[3+x]]), 
{x,2,4}]

does not evaluate, but

IntegrateChangeVariables[int, y, y == x - 3];

Activate[%]

gives the correct answer. Again, mere "direct button pushing" does not work; the user must see and exploit mathematical knowledge.

I want my students to learn these special tricks and insights, and the more examples I have the better.

I think examples here would be of great use to our community (and to WRI as it refines its symbolic algorithms).

Answer 1

$Version

(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)

Clear["Global`*"]

The following integral returns unevaluated

Assuming[x > 0, Integrate[1 - (1 - Exp[-t/x])^x, {t, 0, ∞}]]

Evaluating for specific integer values of x

seq = Integrate[1 - (1 - Exp[-t/#])^#, {t, 0, ∞}] & /@ Range[7]

(* {1, 3, 11/2, 25/3, 137/12, 147/10, 363/20} *)

Use FindSequenceFunction to generalize from the sequence -- to include for non-integer values of x

int[x_] = FindSequenceFunction[seq, x] // FullSimplify

(* x HarmonicNumber[x] *)

Verifying the result by comparing with numerical integration

intN[x_?NumericQ] := NIntegrate[1 - (1 - Exp[-t/x])^x, {t, 0, ∞}]

Plot[{int[x], intN[x]}, {x, 0, 1000},
 PlotStyle -> {Automatic, Dashed},
 PlotLegends -> Placed[{"Exact", "Numeric"}, {.75, .3}],
 AxesLabel -> (Style[#, 14] & /@ {x, int})]