Answer
$$\frac{121}{5}$$
Work Step by Step
\[
\int_{0}^{1}(1+2 x)^{4} d x
\]
Substitute $u=2 x+1$ and $d u =2 d x\Longrightarrow \frac{1}{2} d u=d x$
Substituting $0=x$ in $1+2 x=u$ gives us $1=u$
Substituting $1=x$ in $1+2 x=u$ gives us $3=u$
Limits of integration will change from $\int_{0}^{1}$ to $\int_{1}^{3}$
$=\frac{1}{2} \int_{1}^{3} u^{4} d u$
$=\frac{1}{2}\left[\frac{u^{5}}{5}\right]_{1}^{3}$
$=\left[-\frac{1^{5}}{5}+\frac{3^{5}}{5}\right]\frac{1}{2}$
$=\left[-\frac{1}{5}+\frac{243}{5}\right]\frac{1}{2}$
$=\left[\frac{242}{5}\right]\frac{1}{2}$
$=\frac{121}{5}$
