Using integration by substitution:
u = x^3
du/dx = 3x^2 ---> therefore dx = 1/3x^2 du
Substitute u=x^3 and dx = 1/3x^2du into the original expression to get rid of as many values of x as you can:
∫(x^2) (e^x^3) (dx) =
∫(x^2) (e^u) (1/3x^2 du) =
∫ e^u / 3 du = [both x^2 cancel each other out here so you are left with no more values of x]
1/3 ∫e^u du = [take out a factor of 1/3]
1/3 e^u + c [the integral of e^u is just e^u]
Substitute u = x^3 again to get the answer back in terms of x:
1/3 e^x^3 + c

Reply 5

DFranklin

18

Original post by GeT_iN_SHinJI

~snip~

Please follow the forum guidelines by not posting full solutions:

https://www.thestudentroom.co.uk/showpost.php?p=73536820&postcount=3

Reply 6

username5063840

OP

14

Original post by GeT_iN_SHinJI

Using integration by substitution:
u = x^3
du/dx = 3x^2 ---> therefore dx = 1/3x^2 du
Substitute u=x^3 and dx = 1/3x^2du into the original expression to get rid of as many values of x as you can:
∫(x^2) (e^x^3) (dx) =
∫(x^2) (e^u) (1/3x^2 du) =
∫ e^u / 3 du = [both x^2 cancel each other out here so you are left with no more values of x]
1/3 ∫e^u du = [take out a factor of 1/3]
1/3 e^u + c [the integral of e^u is just e^u]
Substitute u = x^3 again to get the answer back in terms of x:
1/3 e^x^3 + c

oh k thank you.

I tried to work out the answer by parts but couldnt integrate e^3x but this makes more sense