Question

ANYONE IN CALCULUS please help, where does the fraction in the integral even come from? I've tried to wrap my head around it but it doesnt even make sense nor when i look up on google ( where does fraction behind integral sign come from) get no results\. Please define where 1/3 is derived from:

{INTEGRAL SIGN} x^2*e^x^3 = 1/3 {INTEGRAL SIGN} e^x^3*(3x^2) dx = 1/3*e^x^3 + c

Answer 1

Explanation:

You know that the derivative of e^u is e^u·du.

When u = x^3, du = 3x^2·dx. This means x^2·dx = (1/3)·du.

So your integral is ...

`\displaystyle\int{x^2\cdot e^(x^3)}\,dx=\int{(1)/(3)du\cdot e^u}=(1)/(3)\int{e^u}\,du`