Exo 105 (22-SMN2U04-C3-EX105)

$a)$ Calculons $$\int 3(8 y-1) e^{4 y^2-y} d y.$$ On pose $$u=4 y^2-y.$$ On a $$\frac{d u}{d y}=8 y-1$$ et ainsi $$d u=(8 y-1) d y.$$

\begin{aligned}
\int 3(8 y-1) e^{4 y^2-y} d y&=3 \int e^u d u \\ & =3 e^u+C \\
& =3 e^{4 y^2-y}+C
\end{aligned}

$b)$ Calculons$$\int x^2\left(3-10 x^3\right)^4 d x.$$ On pose $$u=3-10 x^3.$$ On a $$\frac{d u}{d x}=-30 x^2$$ et ainsi $$x^2 d x=-\frac{1}{30} d u$$

On a ainsi :

\begin{aligned}
\int x^2\left(3-10 x^3\right)^4 d x & =-\frac{1}{30} \int u^4 d w \\
& =-\frac{1}{30} \cdot \frac{1}{s} u^5+C \\
& =-\frac{1}{150}\left(3-10 x^3\right)^5+C
\end{aligned}

$c)$ Calculons $$\int \frac{x}{\sqrt{1-4 x^2}} d x. $$ On pose $$u=1-4 x^2$$ et l’on a $$d u=-8 x d x.$$

Par conséquent \begin{aligned}
\int \frac{x}{\sqrt{1-4 x^2}} d x & =-\frac{1}{8} \int u^{-\frac{1}{2}} d u \\
& =-\frac{2}{8} u^{\frac{1}{2}}+C \\
& =-\frac{1}{4}\left(1-4 x^2\right)^{1 / 2}+C \\
& =\frac{-\sqrt{1-4 x^2}}{4}
\end{aligned}