Sea la serie \( S=\sum_{k=1}^{\infty} \frac{2^{k}}{k!} \). La serie Ses:

Use the substitution \( u=x^{2}+7 \) to find the following indefinite integral. Check your answer by differentiation. \[ \int 2 x\left(x^{2}+7\right)^{4} d x \] \( \int 2 x\left(x^{2}+7\right)^{4} d x=\frac{\left(x^{2}+7\right)^{5}}{5}+C \)

9/16/24, 1:33 PM 8. Use a change of variables to evaluate the definite integral. \( \int_{0}^{2} x \sqrt{4-x^{2}} d x \) \( \int_{0}^{2} x \sqrt{4-x^{2}} d x= \)

Soi \( f \) une solution non nulle de \( (E): y^{\prime \prime}-\frac{1}{1+t^{2}} y^{\prime}+y=0 \). On note \( q: t \in \mathbb{R}^{+} \mapsto \frac{1}{1+t^{2}} \) 3. Justifier à l'aide du théorème de Cauchy-Lipschitz, que pour tout \( t \in \mathbb{R},\left(f(t), f^{\prime}(t)\right) \neq(0,0) \) 4. En déduire qu'il existe deux applications \( r \in \mathcal{C}^{1}\left(\mathbb{R}^{+}, \mathbb{R}_{+}^{*}\right) \) et \( \theta \in \mathcal{C}^{1}\left(\mathbb{R}^{+}, \mathbb{R}\right) \) telles que \[ \forall t \in \mathbb{R}^{+}, f(t)=r(t) \cos (\theta(t)) \text { et } f^{\prime}(t)=r(t) \sin (\theta(t)) \]

\( \left. \begin{array} { l } { M = x y \cos x y + \sin x y } \\ { \frac { \partial M } { \partial y } = ? } \end{array} \right. \)

10. Evaluate the following integral. \( \int \frac{x^{2}}{x^{3}+2} d x \) \( \left.\int \frac{x^{2}}{x^{3}+2} d x=\frac{1}{3}|n| x+2 \right\rvert\,+C \)