Answer
1
Work Step by Step
The method of example 4, as suggested by the hint, involves substitution.
Remember to express the LIMITS of integration accordingly.
$\left[\begin{array}{ll}
u=-x+1 & du=-dx\\
dx=-du & \\
& \\
x=0\Rightarrow & u=1\\
x=1\Rightarrow & u=0
\end{array}\right]$
$\displaystyle \int_{0}^{1}8(-x+1)^{7}dx=\int_{1}^{0}8u^{7}(-du)$
$\displaystyle=\int_{1}^{0}-8u^{8}du$
... Apply the FTC, def.integral = [antiderivative$]_{a}^{b}$
$=\left[\dfrac{-8u^{8}}{8}\right]_{1}^{0}=\left[-u^{8}\right]_{1}^{0}$
$=0-(-1)=1$
