<prompt> TathminiIint_0infty fracd x4x4 .  <response>: Kamay12 x3 x24 x3ldetsn xn-1ldotstunax yx2 x23 x34 x4ldetsn xnldotsna hivyoy-x y1xx2x3ldetsxn-1ldetsfrac11-xtangu x1. Kwa hiyoyfrac11-x2 .Thamani ya wastani ya x nibarxfracint_0infty x fx d xint_0infty fx d xfracint_0 infty x2 e-x lambda d xint_0infty x e-x lambda d xfrac2 lambda int_0infty x e-x lambda d xint_0infty x e-x lambda d x2 lambda,ambapo ujumuishaji kwa sehemu umetumika kwa nambari na lambda0 imechukuliwa . Thamani inayowezekana zaidi ya x, x_m, imetolewa nakushotofracd td xkulia_x_m0,kutoka x_mlambda.Kumbuka kuwa ikiwa lambda0, fx haina kikomo kwa infty na thamani za wastani na zinazowezekana pia hazina kikomo.Mtini. 3.14AndikaanzailiyopangwaMimi frac12 int_0infty fracd xx44frac12 int_0 infty fracd xx44 frac12 int_-infty0 fracd xx44frac12 int_0 infty fracd xx44frac12 int_inftyinfty fracd xx4 4mwishoiliyopangwana kuzingatiaoint_C fracd zz44ambapo C ni mtaro uliofungwa wa Mtini. 3.14 unaojumuisha mstari kutoka -R hadi R na nusu duara Gamma, iliyopitiwa kwa maana ya kinyume cha saa, katika ndege changamano. Kamafrac1z44frac1leftz-z_1kulialeftz-z_2rightleftz- z_3kuliakushotoz-z_4kulia,wapiz_11i, quad z_2-1i, quad z_3-1-i, quad z_41-itheorem ya mabaki inatoaanzailiyopangwaoint_C fracd zz44 2 pi ikushotofrac1leftz_1-z_2rightleftz_1-z_3rightleftz_ 1-z_4kuliafrac1leftz_2-z_1kulialeftz_2-z_3kulialeft z_2-z_4kuliakulia 2 pi ikushotofrac1222 i cdot 2 i-frac12 cdot 2 i-22 ikulia fracpi4 .mwishoiliyopangwaLakinioint_C fracd zz44int_-RR fracd xx44int_Gamma fracd zz44 mshale wa kulia int_-inftyinfty fracd xx44kama R mshale wa kulia infty. Kwa hiyoIfrac12 cdot fracpi4fracpi8 . 