How to integrate the function in Matlab?

given y(t) = a*x+b*(((t^2/2)+t))+c*(((t^3/3)+t^2+t))+d*(((t^4/4)+t^3+((3*t^2)/2)+t))+e*(((t^5/5)+t^4+2*t^3+2*t^2+t))+f*(((t^6/6)+t^5+((5*t^4)/2)+((10*t^3)/3)+((5*t^2)/2)+t))

int((exp(x-t)*(y(t)^3)),t,0,1)

Answer 1

[Ameer Hamza]

Try this

syms t

s = 0.3;
a = 0.2;
b = 0.5;
c = 0.3;
d = 0.8;
e = 0.1;
f = 0.8;
x = 0.3;

y = a*x+b*(((t^2/2)+t))+c*(((t^3/3)+t^2+t))+d*(((t^4/4)+t^3+((3*t^2)/2)+t))+e*(((t^5/5)+t^4+2*t^3+2*t^2+t))+f*(((t^6/6)+t^5+((5*t^4)/2)+((10*t^3)/3)+((5*t^2)/2)+t));
y_int = int((exp(x-t)*(y^3)),t,0,1);
y_int2 = vpa(y_int);
disp(y_int) % symbolic output
disp(y_int2) % output in decimal format

Result

(3*exp(-7/10)*(14312622602045415832*exp(1) - 38905741936622033629))/1000000
164.22632490929883023849072628867

[Ravikiran Mundewadi]

a,b,c,d,e,f are constants without values how to calculate?

[Ameer Hamza]

If they are unknown constants, then you can also define them as symbolic variables

syms t s a b c d e f x

y = a*x+b*(((t^2/2)+t))+c*(((t^3/3)+t^2+t))+d*(((t^4/4)+t^3+((3*t^2)/2)+t))+e*(((t^5/5)+t^4+2*t^3+2*t^2+t))+f*(((t^6/6)+t^5+((5*t^4)/2)+((10*t^3)/3)+((5*t^2)/2)+t));
y_int = int((exp(x-t)*(y^3)),t,0,1);

The output will be a symbolic expression in term of (s a b c d e f x).

y_int = 20339574*d^3*exp(x) + 28436783046*e^3*exp(x) + 80571537949566*f^3*exp(x) - (4821*b^3*exp(x - 1))/8 - (2665441*c^3*exp(x - 1))/27 - (3538475895*d^3*exp(x - 1))/64 - (9662398822711*e^3*exp(x - 1))/125 - (1752129179993373*f^3*exp(x - 1))/8 + 222*b^3*exp(x) + 36318*c^3*exp(x) - a^3*x^3*exp(x - 1) + 17838*b*c^2*exp(x) + 3252*b^2*c*exp(x) + 914130*b*d^2*exp(x) + 19818*b^2*d*exp(x) + 77530230*b*e^2*exp(x) + 6773016*c*d^2*exp(x) + 143892*b^2*e*exp(x) + 819570*c^2*d*exp(x) + 9819703674*b*f^2*exp(x) + 676668594*c*e^2*exp(x) + 1205850*b^2*f*exp(x) + 7218954*c^2*e*exp(x) + 98703428940*c*f^2*exp(x) + 7108899858*d*e^2*exp(x) + 72227142*c^2*f*exp(x) + 634676832*d^2*e*exp(x) + 1184848286826*d*f^2*exp(x) + 7404854034*d^2*f*exp(x) + 16114285888956*e*f^2*exp(x) + 1137476253630*e^2*f*exp(x) - (96967*b*c^2*exp(x - 1))/2 - (35345*b^2*c*exp(x - 1))/4 - (79515243*b*d^2*exp(x - 1))/32 - (861849*b^2*d*exp(x - 1))/16 - (10537449387*b*e^2*exp(x - 1))/50 - (294575189*c*d^2*exp(x - 1))/16 - (7822617*b^2*e*exp(x - 1))/20 - (26733731*c^2*d*exp(x - 1))/12 - (213541775907*b*f^2*exp(x - 1))/8 - (45984397557*c*e^2*exp(x - 1))/25 - (26222619*b^2*f*exp(x - 1))/8 - (294347011*c^2*e*exp(x - 1))/15 - (1073214948765*c*f^2*exp(x - 1))/4 - (1932399324261*d*e^2*exp(x - 1))/100 - (1178002201*c^2*f*exp(x - 1))/6 - (138018436773*d^2*e*exp(x - 1))/80 - (51532025078445*d*f^2*exp(x - 1))/16 - (644111363199*d^2*f*exp(x - 1))/32 - (876063410205933*e*f^2*exp(x - 1))/20 - (30919810303923*e^2*f*exp(x - 1))/10 + a^3*x^3*exp(x) - (447*a*b^2*x*exp(x - 1))/4 + 6*a^2*b*x^2*exp(x) - (5221*a*c^2*x*exp(x - 1))/3 + 15*a^2*c*x^2*exp(x) - (892563*a*d^2*x*exp(x - 1))/16 + 48*a^2*d*x^2*exp(x) - (80434323*a*e^2*x*exp(x - 1))/25 + 195*a^2*e*x^2*exp(x) - (1179786123*a*f^2*x*exp(x - 1))/4 + 978*a^2*f*x^2*exp(x) + 242628*b*c*d*exp(x) + 1948164*b*c*e*exp(x) + 17901792*b*c*f*exp(x) + 16125516*b*d*e*exp(x) + 161510964*b*d*f*exp(x) + 130095204*c*d*e*exp(x) + 1681817136*b*e*f*exp(x) + 1409661276*c*d*f*exp(x) + 15791037732*c*e*f*exp(x) + 177725405940*d*e*f*exp(x) - (27*a^2*b*x^2*exp(x - 1))/2 - 37*a^2*c*x^2*exp(x - 1) - (501*a^2*d*x^2*exp(x - 1))/4 - (2613*a^2*e*x^2*exp(x - 1))/5 - (5295*a^2*f*x^2*exp(x - 1))/2 - (2638063*b*c*d*exp(x - 1))/4 - 5295635*b*c*e*exp(x - 1) - (97324157*b*c*f*exp(x - 1))/2 - (876673227*b*d*e*exp(x - 1))/20 - (3512258097*b*d*f*exp(x - 1))/8 - (707270753*c*d*e*exp(x - 1))/2 - (45716528721*b*e*f*exp(x - 1))/10 - (15327426191*c*d*f*exp(x - 1))/4 - (214622453951*c*e*f*exp(x - 1))/5 - (9662154824511*d*e*f*exp(x - 1))/20 + 42*a*b^2*x*exp(x) + 642*a*c^2*x*exp(x) + 20526*a*d^2*x*exp(x) + 1183614*a*e^2*x*exp(x) + 108504786*a*f^2*x*exp(x) - 825*a*b*c*x*exp(x - 1) - (15357*a*b*d*x*exp(x - 1))/4 - (108957*a*b*e*x*exp(x - 1))/5 - (36995*a*c*d*x*exp(x - 1))/2 - (292551*a*b*f*x*exp(x - 1))/2 - (593894*a*c*e*x*exp(x - 1))/5 - 891961*a*c*f*x*exp(x - 1) - (8040531*a*d*e*x*exp(x - 1))/10 - (26807721*a*d*f*x*exp(x - 1))/4 - 58988181*a*e*f*x*exp(x - 1) + 306*a*b*c*x*exp(x) + 1416*a*b*d*x*exp(x) + 8022*a*b*e*x*exp(x) + 6810*a*c*d*x*exp(x) + 53820*a*b*f*x*exp(x) + 43704*a*c*e*x*exp(x) + 328146*a*c*f*x*exp(x) + 295806*a*d*e*x*exp(x) + 2465520*a*d*f*x*exp(x) + 21700566*a*e*f*x*exp(x)

you can use subs() function to substitute specific values of these constants