Consider the integral $\displaystyle \int\,3\,x^{3}\,(x^{4} + 1)\,dx$. In the following, we will evaluate the integral using two methods.

A. First, rewrite the integral by multiplying out the integrand:
$\displaystyle \int\,3\,x^{3}\,(x^{4} + 1)\,dx = \int\,$ $\,dx$

Then evaluate the resulting integral term-by-term:
$\displaystyle \int\,3\,x^{3}\,(x^{4} + 1)\,dx =$

B. Next, rewrite the integral using the substitution $w = x^{4} + 1$:
$\displaystyle \int\,3\,x^{3}\,(x^{4} + 1)\,dx = \int\,$ $\,dw$

Evaluate this integral (and back-substitute for $w$) to find the value of the original integral:
$\displaystyle \int\,3\,x^{3}\,(x^{4} + 1)\,dx =$

C. How are your expressions from parts (A) and (B) different? What is the difference between the two? (Ignore the constant of integration.)
(answer from B)$-$(answer from A) =

Are both of the answers correct? (Be sure you can explain why they are!)