Exo 99 (22-SMN2U04-C3-EX99)

$a)$ \begin{aligned} f(x) & =x^2 \\ \int f(x) d x & =\int x^2 d x \\ & =\frac{x^3}{3}+C \end{aligned}

$b)$ \begin{gathered} f(x)=\frac{1}{x} \\ \int f(x) d x=\ln|x|+C \end{gathered}

$c)$ \begin{aligned}
f(x) & =x^{3 / 2} \\
\int f(x) d x & =\frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+C \\
& =\frac{x^{\frac{5}{2}}}{\frac{5}{2}}+C \\
& =\frac{2 x^{\frac{5}{2}}}{5}+C
\end{aligned}

$d)$ \begin{aligned}
& f(x)=4 \\
& \int f(x)=4 x+C
\end{aligned}

$e)$ \begin{aligned}
& f(x)=\cos (x) \\
& \int f(x) d x=\sin (x)+C
\end{aligned}

$f)$ $$f(x)=10^x$$

Rappel : $u^v=\exp (u \ln (v))$ où l’on utilise $\ln \left(v^u\right)=u \ln (v)$.

On a ainsi $$\begin{aligned}
f(x) & =10^x \\
& =\exp (\ln (10) x)
\end{aligned}$$

et \begin{aligned}
\int f(x) d x & =\frac{e^{\ln (10) x}}{\ln (10)} \\
& =\frac{10^x}{\ln (10)}
\end{aligned}