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

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Answer 1 · 2017-09-06T23:25:53

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

#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#

Answer 2 · 2017-09-07T07:35:53

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

#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

ie#d/(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#