Exercice 59

\(\begin{align*} \int x² \sqrt{x+4}.dx \end{align*}\)

Connaissances:

  • Changement de variable
  • Primitives de polynômes

\( \begin{align*} I & = \int x² \sqrt{x+4}.dx \end{align*}\)

Faisons le changement de variable \(u=x+4 \Rightarrow dx=du\)

\(\begin{align*}I & = \int x² \sqrt{x+4}.dx \\
& = \int (u-4)² \sqrt u .du \\
& = \int (u²-8u+16)\sqrt u .du \\
& = \int \big[u^{5/2}-8u^{3/2} + 16u^{1/2} \big] du \\
& = \frac{2}{7}u^{7/2}-8 \frac{2}{5}u^{5/2}+16 \frac{2}{3}u^{3/2} +C \\
& = \frac{2}{7}(x+4)^{7/2}- \frac{16}{5}(x+4)^{5/2}+\frac{32}{3}(x+4)^{3/2} +C \\
\end{align*}\)

 

\[\boxed {\begin{align*} I =\frac{2}{7}(x+4)^{7/2}- \frac{16}{5}(x+4)^{5/2}+\frac{32}{3}(x+4)^{3/2} +C (\in \mathbb R) \end{align*}}\]