Answer
$4095$
Work Step by Step
The method of example4, as suggested by the hint, involves substitution.
Remember to express the LIMITS of integration accordingly.
$\left[\begin{array}{ll}
u=3x+1 & du=3dx\\
dx=du/3 & \\
& \\
x=0\Rightarrow & u=1\\
x=1\Rightarrow & u=4
\end{array}\right]$
$\displaystyle \int_{0}^{1}18(3x+1)^5dx=\int_{1}^{4}18u^{5}(\frac{du}{3})=\int_{1}^{4}6u^{5}du$
... Apply the FTC, (def.integral = [antiderivative$]_{a}^{b}$)
$=\left[\dfrac{6u^{6}}{6}\right]_{1}^{4}=\left[u^{6}\right]_{1}^{4}$
$=4^{6}-1^{6}$
$=4095$
