Exercice 30

\(\begin{align*}\int^{5}_3 (x-3)^9dx \end{align*}\)

\(\begin{align*}I = \int^{5}_3 (x-3)^9dx \end{align*}\)

Faisons un simple changement de variable: \(u=x-3\)

\(\begin{align*}
& \begin{cases} u=x-3 \\ du =dx \end{cases} \Rightarrow
\begin{cases} x=3 \Rightarrow u =0 \\ x=5 \Rightarrow u=2  \end{cases} \\
I & = \int^{x=5}_{x=3} (x-3)^9dx \\
& = \int^{u=2}_{u=0} (u)^9.du \\
& = \big[\frac{1}{10}u^{10} \big]_0^2  \\
& = \frac{1}{10}2^{10} - \cancel{ \frac{1}{10}0^{10}} \\
& = \frac{1024}{10}= \frac{512}{5}
\end{align*}\)

\[\begin{align*} \boxed{I= \frac{512}{5} }\end{align*}\]