# Different expressions for an integral if I split the region of integration

Consider the assumptions

$Assumptions = 1 > v > 0 && y1 \[Element] Reals && y2 \[Element] Reals && k > 0 && 1 > t1 > 0  and define y[t_, y1_, y2_, t1_] = Piecewise[{{y1, 0 <= t <= t1}, {y2, t1 < t <= 1}}, 0];  and two different formulations of the same integral, Assuming[t1 + v < 1, Integrate[ Assuming[{0 <= t <= 1, t1 + v < 1}, Integrate[(1 + Exp[-2 Integrate[y[s, y1, y2, t1]^2, {s, t - u - v, t - v}]])/ 2, {u, 0, t}]], {t, v, t1 + v}]] // Simplify  and Assuming[t1 + v < 1, Integrate[ Assuming[{0 <= t <= 1, t1 + v < 1}, Integrate[(1 + Exp[-2 Integrate[ y[s, y1, y2, t1]^2, {s, t - u - v, t - v}]])/2, {u, 0, t - v}]], {t, v, t1 + v}]] + Assuming[t1 + v < 1, Integrate[ Assuming[{0 <= t <= 1, t1 + v < 1}, Integrate[(1 + Exp[-2 Integrate[ y[s, y1, y2, t1]^2, {s, t - u - v, t - v}]])/2, {u, t - v, t}]], {t, v, t1 + v}]] // Simplify  The two formulations differ for the region of integration, that in the second case is splitted in two parts, because the interval in u between 0 and t is splitted in [0,t-v]+[t-v,t],and then the integrals summed. Why Mathematica give me back two different expressions? - Why all those separate Assuming functions? Can't you combine them? – Sjoerd C. de Vries Jan 16 '13 at 20:11 The D you use as a variable is a reserved word. – Sjoerd C. de Vries Jan 16 '13 at 20:13 I combine a bit Assuming. I changed the variable from D to u (in my code I use greek symbols, but to be more clear I changed in D the variable symbol without checking). Also I've added some necessary$Assumptions. The main point is still there. – Nicola Jan 16 '13 at 21:07
You have a k>0 which seems not needed. – b.gatessucks Jan 17 '13 at 9:48

Maybe not the answer you're looking for. It took a long time to run your code as it is so I changed it a bit. The bottom line is that Mathematica is doing fine and the two expressions you get just look different.

Some definitions :

myAssum = 1 > v > 0 && y1 \[Element] Reals && y2 \[Element] Reals && 1 > t1 > 0
myConditions = {tMv -> t - v};


because until the last integration everything depends on t-v only, so no point carrying two variables.

intY2 = Simplify[Integrate[PiecewiseExpand[y[s, y1, y2, t1]^2], {s, tMv - u, tMv},
Assumptions -> {0 <= tMv <= t1, 0 <= u <= t}],
Assumptions -> {myAssum, 0 <= tMv <= t1, 0 <= u <= t}];

intExp = Integrate[(1 + Exp[-2 intY2])/2, {u, 0, t},
Assumptions -> {0 <= tMv <= t1, 0 <= t <= 2}];


Now we substitute back tMv -> t-v and do the last integration :

result = Integrate[intExp /. myConditions, {t, v, t1 + v},
Assumptions -> myAssum];


To check the other approach I just repeat the last two steps :

intExp1 = Integrate[(1 + Exp[-2 intY2])/2, {u, 0, tMv},
Assumptions -> {0 <= tMv <= t1, 0 <= t <= 2}];
intExp2 = Integrate[(1 + Exp[-2 intY2])/2, {u, tMv, t},
Assumptions -> {0 <= tMv <= t1, 0 <= t <= 2}];

result1 = Integrate[intExp1 /. myConditions, {t, v, t1 + v},
Assumptions -> myAssum];
result2 = Integrate[intExp2 /. myConditions, {t, v, t1 + v},
Assumptions -> myAssum];


Check :

(result == result1 + result2 ) // Simplify
(* True *)

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