Tue Mar 10 10:29:23 MST 1998 euler% math Mathematica 3.0 for Solaris Copyright 1988-96 Wolfram Research, Inc. -- Terminal graphics initialized -- In[1]:= In[2]:= In[3]:= (* ----------[ M a t h e m a t i c a ]---------- *) 0. Second In[4]:= (* ---------- Initialization ---------- *) 0. Second In[5]:= (* ---------- Definite Integrals ---------- *) 0. Second In[6]:= (* The following two functions have a pole at a. The first integral\ > has a principal value of zero; the second is divergent *) 0. Second In[7]:= Integrate[1/(x - a), {x, a - 1, a + 1}] 1 Integrate::idiv: Integral of ------ does not converge on {-1 + a, 1 + a}. -a + x 0.65 Second 1 Out[7]= Integrate[------, {x, -1 + a, 1 + a}] -a + x In[8]:= Integrate[1/(x - a), {x, a - 1, a + 1}, PrincipalValue -> True] 1.59 Second 1 Out[8]= If[a > 1, 0, Integrate[------, {x, -1 + a, 1 + a}]] -a + x In[9]:= Integrate[1/(x - a)^2, {x, a - 1, a + 1}] 0.12 Second Out[9]= Infinity In[10]:= (* Different branches of the square root need to be chosen in the\ > intervals [0, 1] and [1, 2]. The correct results are 4/3, [4 - sqrt(8)]/3, [8 - sqrt(8)]/3, respectively *) 0. Second In[11]:= Integrate[Sqrt[x + 1/x - 2], {x, 0, 1}] 0.8 Second 4 Out[11]= - 3 In[12]:= Integrate[Sqrt[x + 1/x - 2], {x, 1, 2}] 0.62 Second 4 2 Sqrt[2] Out[12]= - - --------- 3 3 In[13]:= Integrate[Sqrt[x + 1/x - 2], {x, 0, 2}] 1.13 Second 8 2 Sqrt[2] Out[13]= - - --------- 3 3 In[14]:= (* => sqrt(2) [a modification of a problem due to W. Kahan] *) 0. Second In[15]:= Integrate[Sqrt[2 - 2*Cos[2*x]]/2, {x, -3*Pi/4, -Pi/4}] 0.56 Second Out[15]= Sqrt[2] In[16]:= (* Contour integrals => pi/a e^(-a) for a > 0. See Norman Levinson\ > and Raymond M. Redheffer, _Complex Variables_, Holden-Day, Inc., 1970, p. 198. *) 0. Second In[17]:= Integrate[Cos[x]/(x^2 + a^2), {x, -Infinity, Infinity}, Assumptions -> a > 0] 0.6 Second Pi Out[17]= ---- a a E In[18]:= (* Integrand with a branch point => pi/sin(pi a) for 0 < a < 1 [Levinson and Redheffer, p. 212] *) 0.01 Second In[19]:= Integrate[t^(a - 1)/(1 + t), {t, 0, Infinity}, Assumptions -> 0 < a\ > < 1] 0.51 Second Out[19]= Pi Csc[a Pi] In[20]:= (* Integrand with a residue at infinity => -2 pi [sin(pi/5) + sin(2\ > pi/5)] (principal value) [Levinson and Redheffer, p. 234] *) 0. Second In[21]:= Integrate[5*x^3/(1 + x + x^2 + x^3 + x^4), {x, -Infinity, Infinity}, PrincipalValue -> True] Integrate::idiv: 3 5 x Integral of -------------------- does not converge on {-Infinity, 2 3 4 1 + x + x + x + x Infinity}. Integrate::idiv: 3 x Integral of -------------------- does not converge on {-Infinity, 2 3 4 1 + x + x + x + x Infinity}. General::stop: Further output of Integrate::idiv will be suppressed during this calculation. 0.19 Second 3 x Out[21]= 5 Integrate[--------------------, {x, -Infinity, Infinity}, 2 3 4 1 + x + x + x + x > PrincipalValue -> True] In[22]:= (* integrate(1/[1 + x + x^2 + ... + x^(2 n)], x =\ > -infinity..infinity) = 2 pi/(2 n + 1) [1 + cos(pi/[2 n + 1])] csc(2 pi/[2 n + 1]) [Levinson and Redheffer, p. 255] => 2 pi/5 [1 + cos(pi/5)] csc(2 pi/5) *) 0. Second In[23]:= Integrate[1/(1 + x + x^2 + x^4), {x, -Infinity, Infinity}] 0.55 Second 1 Out[23]= Integrate[---------------, {x, -Infinity, Infinity}] 2 4 1 + x + x + x In[24]:= (* Integrand with a residue at infinity and a branch cut => pi\ > [sqrt(2) - 1] [Levinson and Redheffer, p. 234] *) 0. Second In[25]:= Integrate[Sqrt[1 - x^2]/(1 + x^2), {x, -1, 1}] 1 Power::infy: Infinite expression - encountered. 0 1 Power::infy: Infinite expression - encountered. 0 3.3 Second (-1 + Sqrt[2]) Pi -Pi + Sqrt[2] Pi Out[25]= ----------------- + ---------------- 2 2 In[26]:= Simplify[%] 0.02 Second Out[26]= (-1 + Sqrt[2]) Pi In[27]:= (* This is a common integral in many physics calculations => q/p sqrt(pi/p) e^(q^2/p) (Re p > 0) [Gradshteyn and Ryzhik\ > 3.462(6)] *) 0. Second In[28]:= Integrate[x*Exp[-p*x^2 + 2*q*x], {x, -Infinity, Infinity}] 1.05 Second 2 q /p E Sqrt[Pi] q Out[28]= If[Re[p] > 0 && Re[q] < 0, ----------------, 3/2 p 2 2 q x - p x > Integrate[E x, {x, -Infinity, Infinity}]] In[29]:= (* => 2 Euler's_constant [Gradshteyn and Ryzhik 8.367(5-6)] *) 0. Second In[30]:= Integrate[1/Log[t] + 1/(1 - t) - Log[Log[1/t]], {t, 0, 1}] 5.33 Second 1 1 1 Out[30]= Integrate[----- + ------ - Log[Log[-]], {t, 0, 1}] 1 - t Log[t] t In[31]:= (* This integral comes from atomic collision theory => 0 [John\ > Prentice] *) 0. Second In[32]:= Integrate[Sin[t]/t*Exp[2*I*t], {t, -Infinity, Infinity}] 1.35 Second Out[32]= 0 In[33]:= (* => 1/12 [Gradshteyn and Ryzhik 6.443(3)] *) 0. Second In[34]:= Integrate[Log[Gamma[x]]*Cos[6*Pi*x], {x, 0, 1}] 18.01 Second Out[34]= Integrate[Cos[6 Pi x] Log[Gamma[x]], {x, 0, 1}] In[35]:= (* => 36/35 [Gradshteyn and Ryzhik 7.222(2)] *) 0. Second In[36]:= Integrate[(1 + x)^3*LegendreP[1, x]*LegendreP[2, x], {x, -1, 1}] 0.03 Second 36 Out[36]= -- 35 In[37]:= (* => 1/sqrt(a^2 + b^2) (a > 0 and b real) [Gradshteyn and Ryzhik 6.611(1)] *) 0. Second In[38]:= Integrate[Exp[-a*x]*BesselJ[0, b*x], {x, 0, Infinity}] 0.52 Second 1 Out[38]= If[Im[b] == 0 && Re[a] > 0, -------------, 2 2 Sqrt[a + b ] BesselJ[0, b x] > Integrate[---------------, {x, 0, Infinity}]] a x E In[39]:= (* Integrand contains a special function => 4/(3 pi) [Tom\ > Hagstrom] *) 0. Second In[40]:= Integrate[(BesselJ[1, x]/x)^2, {x, 0, Infinity}] 0.24 Second 4 Out[40]= ---- 3 Pi In[41]:= (* => (cos 7 - 1)/7 [Gradshteyn and Ryzhik 6.782(3)] *) 0. Second In[42]:= Integrate[CosIntegral[x]*BesselJ[0, 2*Sqrt[7*x]], {x, 0, Infinity}] 0.53 Second Out[42]= Integrate[BesselJ[0, 2 Sqrt[7] Sqrt[x]] CosIntegral[x], > {x, 0, Infinity}] In[43]:= (* This integral comes from doing a two loop Feynman diagram for a\ > QCD problem => - [17/3 + pi^2]/36 + log 2/9 [35/3 - pi^2/2 - 4 log 2 + log(2)^2] + zeta(3)/4 = 0.210883... [Rolf Mertig] *) 0. Second In[44]:= Integrate[x^2*PolyLog[3, 1/(x + 1)], {x, 0, 1}] 0.22 Second 2 1 Out[44]= Integrate[x PolyLog[3, -----], {x, 0, 1}] 1 + x In[45]:= NIntegrate[x^2*PolyLog[3, 1/(x + 1)], {x, 0, 1}] 0.01 Second Out[45]= 0.210883 In[46]:= N[- (17/3 + Pi^2)/36 + Log[2]/9*(35/3 - Pi^2/2 - 4*Log[2] + Log[2]^2) + Zeta[3]/4] 0.01 Second Out[46]= 0.210883 In[47]:= (* Integrate a piecewise defined step function s(t) multiplied by\ > cos t, where s(t) = 0 (t < 1); 1 (1 <= t <= 2); 0 (t > 2) => 0 (u < 1); sin u - sin 1 (1 <= u <= 2); sin 2 - sin 1 (u > 2) *) 0. Second In[48]:= s[t_]:= If[1 <= t <= 2, 1, 0]; 0. Second In[49]:= Integrate[s[t]*Cos[t], {t, 0, u}] 0.14 Second Out[49]= Integrate[Cos[t] If[1 <= t <= 2, 1, 0], {t, 0, u}] In[50]:= << Calculus`DiracDelta` 0.47 Second In[51]:= s[t_]:= UnitStep[t - 1] - UnitStep[t - 2]; 0. Second In[52]:= Integrate[s[t]*Cos[t], {t, 0, u}] 0.81 Second Out[52]= (Sin[2] - Sin[u]) UnitStep[-2 + u] + > (-Sin[1] + Sin[u]) UnitStep[-1 + u] In[53]:= Clear[s] 0. Second In[54]:= (* Integrating first with respect to y and then x is much easier than integrating first with respect to x and then y => (|b| - |a|) pi [W. Kahan] *) 0. Second In[55]:= integrate[e_, limits_, assumptions___]:= If[Head[e] =!= If, Integrate[e, limits, assumptions], Module[{I1 = Integrate[e[[2]], limits, assumptions], I2 = Integrate[e[[3]], limits, assumptions]}, IF[e[[1]], I1, I2]]] 0. Second In[56]:= (* Note: e[[3]] evaluates to e in the above which is a bug! *) 0. Second In[57]:= Integrate[Integrate[x/(x^2 + y^2), {y, -Infinity, Infinity}], {x, a,\ > b}, Assumptions -> {a > 0, b > 0}] 3.42 Second 2 -2 Out[57]= Integrate[If[Arg[x ] != Pi, Pi Sqrt[x ] x, x > Integrate[-------, {y, -Infinity, Infinity}]], {x, a, b}, 2 2 x + y > Assumptions -> {a > 0, b > 0}] In[58]:= Integrate[Integrate[x/(x^2 + y^2), {x, a, b}], {y, -Infinity,\ > Infinity}, Assumptions -> {a > 0, b > 0}] 0.88 Second 2 2 2 2 -Log[a + y ] Log[b + y ] Out[58]= Integrate[------------- + ------------, {y, -Infinity, Infinity}, 2 2 > Assumptions -> {a > 0, b > 0}] In[59]:= Integrate[Integrate[x/(x^2 + y^2), {y, -Infinity, Infinity}], {x, a,\ > b}, Assumptions -> {a < 0, b > 0}] 3.43 Second 2 -2 Out[59]= Integrate[If[Arg[x ] != Pi, Pi Sqrt[x ] x, x > Integrate[-------, {y, -Infinity, Infinity}]], {x, a, b}, 2 2 x + y > Assumptions -> {a < 0, b > 0}] In[60]:= Integrate[Integrate[x/(x^2 + y^2), {x, a, b}], {y, -Infinity,\ > Infinity}, Assumptions -> {a < 0, b > 0}] 0.86 Second 2 2 2 2 -Log[a + y ] Log[b + y ] Out[60]= Integrate[------------- + ------------, {y, -Infinity, Infinity}, 2 2 > Assumptions -> {a < 0, b > 0}] In[61]:= Integrate[Integrate[x/(x^2 + y^2), {y, -Infinity, Infinity}], {x, a,\ > b}, Assumptions -> {a < 0, b < 0}] 3.38 Second 2 -2 Out[61]= Integrate[If[Arg[x ] != Pi, Pi Sqrt[x ] x, x > Integrate[-------, {y, -Infinity, Infinity}]], {x, a, b}, 2 2 x + y > Assumptions -> {a < 0, b < 0}] In[62]:= Integrate[Integrate[x/(x^2 + y^2), {x, a, b}], {y, -Infinity,\ > Infinity}, Assumptions -> {a < 0, b < 0}] 0.89 Second 2 2 2 2 -Log[a + y ] Log[b + y ] Out[62]= Integrate[------------- + ------------, {y, -Infinity, Infinity}, 2 2 > Assumptions -> {a < 0, b < 0}] In[63]:= integrate[Integrate[x/(x^2 + y^2), {y, -Infinity, Infinity}], {x, a,\ > b}, Assumptions -> {a > 0, b > 0}] Integrate::gener: Unable to check convergence 4.23 Second 2 Pi Pi Out[63]= IF[Arg[x ] != Pi, -(---------) + ---------, -2 -2 Sqrt[a ] Sqrt[b ] 2 -2 > Integrate[If[Arg[x ] != Pi, Pi Sqrt[x ] x, x > Integrate[-------, {y, -Infinity, Infinity}]], {x, a, b}, 2 2 x + y > Assumptions -> {a > 0, b > 0}]] In[64]:= integrate[Integrate[x/(x^2 + y^2), {y, -Infinity, Infinity}], {x, a,\ > b}, Assumptions -> {a < 0, b > 0}] Integrate::gener: Unable to check convergence 4.28 Second 2 Pi Pi Out[64]= IF[Arg[x ] != Pi, -(---------) + ---------, -2 -2 Sqrt[a ] Sqrt[b ] 2 -2 > Integrate[If[Arg[x ] != Pi, Pi Sqrt[x ] x, x > Integrate[-------, {y, -Infinity, Infinity}]], {x, a, b}, 2 2 x + y > Assumptions -> {a < 0, b > 0}]] In[65]:= integrate[Integrate[x/(x^2 + y^2), {y, -Infinity, Infinity}], {x, a,\ > b}, Assumptions -> {a < 0, b < 0}] Integrate::gener: Unable to check convergence General::stop: Further output of Integrate::gener will be suppressed during this calculation. 4.34 Second 2 Pi Pi Out[65]= IF[Arg[x ] != Pi, -(---------) + ---------, -2 -2 Sqrt[a ] Sqrt[b ] 2 -2 > Integrate[If[Arg[x ] != Pi, Pi Sqrt[x ] x, x > Integrate[-------, {y, -Infinity, Infinity}]], {x, a, b}, 2 2 x + y > Assumptions -> {a < 0, b < 0}]] In[66]:= (* => [log(sqrt(2) + 1) + sqrt(2)]/3 [Caviness et all, section\ > 2.10.1] *) 0. Second In[67]:= Integrate[Integrate[Sqrt[x^2 + y^2], {x, 0, 1}], {y, 0, 1}] 2.6 Second 2 Sqrt[2] + ArcSinh[1] + Log[1 + Sqrt[2]] Out[67]= ----------------------------------------- 6 In[68]:= FullSimplify[%] 0.72 Second Sqrt[2] + ArcSinh[1] Out[68]= -------------------- 3 In[69]:= (* => (pi a)/2 [Gradshteyn and Ryzhik 4.621(1)] *) 0. Second In[70]:= Integrate[Integrate[Sin[a]*Sin[y]/Sqrt[1 -\ > Sin[a]^2*Sin[x]^2*Sin[y]^2], {x, 0, Pi/2}], {y, 0, Pi/2}] 1 Power::infy: Infinite expression - encountered. 0 General::stop: Further output of Power::infy will be suppressed during this calculation. 2007.57 Second Out[70]= -2 I Sqrt[2] Integrate[(Sqrt[1 + > (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y] - 2 > Sqrt[-4 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) ]) / 2] > Sqrt[1 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y] + 2 > Sqrt[-4 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) ]) / 2] > EllipticF[I ArcSinh[Sqrt[1 + Cos[2 a] + Cos[2 y] - > Cos[2 a] Cos[2 y] + 2 > Sqrt[-4 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) ]] / > Sqrt[2]], -1 + ((1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) > (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y] - 2 > Sqrt[-4 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) ])) / > 2] Sin[y]) / > (Sqrt[3 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]] > Sqrt[1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y] + 2 > Sqrt[-4 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) ]]), Pi > {y, 0, --}] Sin[a] 2 In[71]:= Integrate[Sin[a]*Sin[y]/Sqrt[1 - Sin[a]^2*Sin[x]^2*Sin[y]^2], {x, 0,\ > Pi/2}] 12.66 Second Out[71]= (-2 I Sqrt[2] Sqrt[1 + > (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y] - 2 > Sqrt[-4 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) ]) / 2] > Sqrt[1 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y] + 2 > Sqrt[-4 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) ]) / 2] > EllipticF[I ArcSinh[Sqrt[1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y] + 2 > Sqrt[-4 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) ]] / > Sqrt[2]], -1 + ((1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) > (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y] - 2 > Sqrt[-4 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) ])) / 2] > Sin[a] Sin[y]) / > (Sqrt[3 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]] > Sqrt[1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y] + 2 > Sqrt[-4 + (1 + Cos[2 a] + Cos[2 y] - Cos[2 a] Cos[2 y]) ]]) In[72]:= Simplify[%] 33.98 Second Out[72]= (-I Sqrt[2] EllipticF[I 2 > ArcSinh[Sqrt[1 + Cos[2 y] + 2 Cos[2 a] Sin[y] + 2 2 2 > Sqrt[-4 + 4 (Cos[y] + Cos[2 a] Sin[y] ) ]] / Sqrt[2]], 2 2 > -1 + (Cos[y] + Cos[2 a] Sin[y] ) 2 > (1 + Cos[2 y] + 2 Cos[2 a] Sin[y] - 2 2 2 > Sqrt[-4 + 4 (Cos[y] + Cos[2 a] Sin[y] ) ])] Sin[a] Sin[y] 2 > Sqrt[3 + Cos[2 y] + 2 Cos[2 a] Sin[y] - 2 2 2 > Sqrt[-4 + 4 (Cos[y] + Cos[2 a] Sin[y] ) ]]) / 2 > (Sqrt[3 + Cos[2 y] + 2 Cos[2 a] Sin[y] ] 2 > Sqrt[(1 + Cos[2 y] + 2 Cos[2 a] Sin[y] + 2 2 2 > Sqrt[-4 + 4 (Cos[y] + Cos[2 a] Sin[y] ) ]) / 2 > (3 + Cos[2 y] + 2 Cos[2 a] Sin[y] + 2 2 2 > Sqrt[-4 + 4 (Cos[y] + Cos[2 a] Sin[y] ) ])]) In[73]:= Integrate[%, {y, 0, Pi/2}] 0. Second Out[73]= -I Sqrt[2] Integrate[(EllipticF[I 2 > ArcSinh[Sqrt[1 + Cos[2 y] + 2 Cos[2 a] Sin[y] + 2 2 2 > Sqrt[-4 + 4 (Cos[y] + Cos[2 a] Sin[y] ) ]] / Sqrt[2]], 2 2 > -1 + (Cos[y] + Cos[2 a] Sin[y] ) 2 > (1 + Cos[2 y] + 2 Cos[2 a] Sin[y] - 2 2 2 > Sqrt[-4 + 4 (Cos[y] + Cos[2 a] Sin[y] ) ])] Sin[y] 2 > Sqrt[3 + Cos[2 y] + 2 Cos[2 a] Sin[y] - 2 2 2 > Sqrt[-4 + 4 (Cos[y] + Cos[2 a] Sin[y] ) ]]) / 2 > (Sqrt[3 + Cos[2 y] + 2 Cos[2 a] Sin[y] ] 2 > Sqrt[(1 + Cos[2 y] + 2 Cos[2 a] Sin[y] + 2 2 2 > Sqrt[-4 + 4 (Cos[y] + Cos[2 a] Sin[y] ) ]) / 2 > (3 + Cos[2 y] + 2 Cos[2 a] Sin[y] + 2 2 2 Pi > Sqrt[-4 + 4 (Cos[y] + Cos[2 a] Sin[y] ) ])]), {y, 0, --}] Sin[a] 2 In[74]:= (* => 46/15 [Paul Zimmermann] *) 0. Second In[75]:= Integrate[Integrate[Abs[y - x^2], {y, 0, 2}], {x, -1, 1}] 16.08 Second 2 2 2 2 2 2 Out[75]= Integrate[-2 Im[x] Log[Im[x] - Re[x] + Sqrt[(Im[x] + Re[x] ) ]] 2 2 2 2 > Re[x] + 2 Im[x] Log[2 + Im[x] - Re[x] + 4 2 2 2 2 2 > Sqrt[Im[x] + (-2 + Re[x] ) + 2 Im[x] (2 + Re[x] )]] Re[x] - 2 2 2 2 2 > ((Im[x] - Re[x] ) Sqrt[(Im[x] + Re[x] ) ]) / 2 + 2 2 4 2 2 > ((2 + Im[x] - Re[x] ) Sqrt[Im[x] + (-2 + Re[x] ) + 2 2 > 2 Im[x] (2 + Re[x] )]) / 2, {x, -1, 1}] In[76]:= Integrate[Integrate[Abs[y - x^2], {y, 0, 2}, Assumptions -> -1 <= x\ > <= 1], {x, -1, 1}] 2 General::ivar: -x + y is not a valid variable. 2 General::ivar: -x + y is not a valid variable. 2 General::ivar: -x + y is not a valid variable. General::stop: Further output of General::ivar will be suppressed during this calculation. 18.38 Second 2 Out[76]= Integrate[Integrate[Abs[-x + y], {y, 0, 2}, > Assumptions -> -1 <= x <= 1], {x, -1, 1}] In[77]:= Integrate[Abs[y - x^2], {y, 0, 2}] 6.9 Second 2 2 2 2 2 2 2 Out[77]= -2 Im[x] Log[Im[x] - Re[x] + Sqrt[(Im[x] + Re[x] ) ]] Re[x] + 2 2 2 > 2 Im[x] Log[2 + Im[x] - Re[x] + 4 2 2 2 2 2 > Sqrt[Im[x] + (-2 + Re[x] ) + 2 Im[x] (2 + Re[x] )]] Re[x] - 2 2 2 2 2 (Im[x] - Re[x] ) Sqrt[(Im[x] + Re[x] ) ] > ------------------------------------------ + 2 2 2 4 2 2 (2 + Im[x] - Re[x] ) Sqrt[Im[x] + (-2 + Re[x] ) + 2 2 > 2 Im[x] (2 + Re[x] )] > --------------------------------------------------------------------------- 2 In[78]:= Integrate[% /. {Re[x] -> x, Im[x] -> 0}, {x, -1, 1}] 0.82 Second 46 Out[78]= -- 15 In[79]:= (* Multiple integrals: volume of a tetrahedron => a b c / 6 *) 0. Second In[80]:= Integrate[Integrate[Integrate[1, {z, 0, c*(1 - x/a - y/b)}], {y, 0, b*(1 - x/a)}], {x, 0, a}] 0.1 Second a b c Out[80]= ----- 6 In[81]:= (* ---------- Quit ---------- *) 0. Second In[82]:= Quit[] real 5208.66 user 4546.38 sys 1.06