If \(y = x(x^4 + x + 1)\), evaluate \(\int \limits_{0} ^{1} y \mathrm d x\).
A
\(\frac{11}{12}\)
B
1
C
\(\frac{5}{6}\)
D
zero
Verified Correct Answer: B.
Step-by-Step Mathematical Solution
\(y = x(x^{4} + x + 1) = x^{5} + x^{2} + x\)
\(\int \limits_{0} ^{1} (x^{5} + x^{2} + x) \mathrm d x = \frac{x^{6}}{6} + \frac{x^{3}}{3} + \frac{x^{2}}{2}\)
= \([\frac{x^{6}}{6} + \frac{x^{3}}{3} + \frac{x^{2}}{2}]_{0} ^{1}\)
= \(\frac{1}{6} + \frac{1}{3} + \frac{1}{2}\)
= \(1\)
Source: Myschool Past Questions Archive (Record #30857)