How do you integrate intx^2e^(x^3)dx?

2 Answers
Sep 6, 2017

int x^2 e^(x^3) dx = 1/3 e^(x^3) + C

Explanation:

int x^2 e^(x^3) dx

Do a u-substitution:
u=x^3
(du)/(dx) = 3x^2
dx = (du)/(3x^2)

Substitute:
int color(red)(x^2) e^u (du)/(3color(red)(x^2))

= 1/3 int e^u du

= 1/3e^u + C

Now re-substitute in terms of x:
= 1/3 e^(x^3) + C

Sep 7, 2017

intx^2e^(x^3)dx=1/3e^(x^3)+C

Explanation:

intx^2e^(x^3)dx

if we recognise that in general

color(blue)(d/(dx)(e^(f(x)))=f'(x)e^(f(x)))

then noting that in the integral the function in front of the exponential function is a multiple of the derivative of the power

ied/(dx)(e^(x^3))=3x^2e^(x^3)

we can write down the integral by inspection

intx^2e^(x^3)dx=1/3e^(x^3)+C