2πa∫0∞t3e−st−4ta2 dt=e−as
I will use a wrong method and it will help me to get the above result.
x=t
dx=2t1 dt
2πa∫0∞x3e−sx2−4x2a2 2x dx=πa∫0∞x2e−sx2−4x2a2 dx
=πae−as∫0∞x2e−(sx−2xa)2 dx
u=sx−2xa
du=(s+2x2a) dx
πae−as∫−∞∞x2(s+2x2a)e−u2 du=πae−as∫−∞∞(sx2+2a)e−u2 du
Look above, we still have x2 inside the integral.
ux=sx2−2a
x2=sux+2a
Watch the magic now. Let u=0 above. (At here the wrong started.)
x2=2sa
Substitute this value back in the integral.
πae−as∫−∞∞(s[2sa]+2a)e−u2 du=πae−as∫−∞∞(2a+2a)e−u2 du
=πae−as∫−∞∞ae−u2 du=π1e−as∫−∞∞e−u2 du
=21e−as∫−∞∞π2e−u2 du=21e−aserf(u)∣∣∣∣∣−∞∞=21e−as(1+1)=e−as
How could this wrong method give the correct result of the integral? Is it magic or there is an explanation to fill up these mysterious gaps? It worked two times and if no one did it before, Mario will take the credit.
I will use a wrong method and it will help me to get the above result.
x=t
dx=2t1 dt
2πa∫0∞x3e−sx2−4x2a2 2x dx=πa∫0∞x2e−sx2−4x2a2 dx
=πae−as∫0∞x2e−(sx−2xa)2 dx
u=sx−2xa
du=(s+2x2a) dx
πae−as∫−∞∞x2(s+2x2a)e−u2 du=πae−as∫−∞∞(sx2+2a)e−u2 du
Look above, we still have x2 inside the integral.
ux=sx2−2a
x2=sux+2a
Watch the magic now. Let u=0 above. (At here the wrong started.)
x2=2sa
Substitute this value back in the integral.
πae−as∫−∞∞(s[2sa]+2a)e−u2 du=πae−as∫−∞∞(2a+2a)e−u2 du
=πae−as∫−∞∞ae−u2 du=π1e−as∫−∞∞e−u2 du
=21e−as∫−∞∞π2e−u2 du=21e−aserf(u)∣∣∣∣∣−∞∞=21e−as(1+1)=e−as
How could this wrong method give the correct result of the integral? Is it magic or there is an explanation to fill up these mysterious gaps? It worked two times and if no one did it before, Mario will take the credit.