Question

Integration- Someone clarify for me

Answer 1

\[\int\limits_ {}{} x^{3} e^{x^4}\]
For this one, the answer is
\[\frac {1}{4} e^{x^{4}}\]
Could someone work it out with intermediate steps? (Or maybe just clarify how the integral should be simplified

Answer 2

Are you familiar with the concept of U-substitution? :)

Answer 3

\[\large \color{orangered}{u=x^4}\]Taking the derivative of u, with respect to x, gives us,\[\large \frac{du}{dx}=4x^3\]There is a process that allows us to write the dx on the other side.
For our purposes simply think of this as multiplication.
So multiplying both sides by dx gives us,\[\large du=4x^3 dx\]Dividing both sides by 4 gives us,\[\large \color{orangered}{\frac{1}{4}du=x^3dx}\]

Look at the orange pieces.
They should replace some things in your integral very nicely.

Answer 4

See what we did to simplify the integral?
Take a few moments to soak it all in :O it might take an easier problem to really grasp the concept if you haven't seen a substitution before.