Exo 101 (22-SMN2U04-C3-EX101)

$a)$ \begin{aligned} \int 5 t^3-10 t^{-6}+4 d t&=\int 5 t^3 d t-\int 10 t^{-6} d t+\int 4 d t \\
&=5 \int t^3 d t-10 \int t^{-6} d t+4 \int d t\\
&= \frac{5t^4}{4}-10 \frac{t^{-5}}{-5}+4 t+C \\
&= \frac{5t^4}{4}+2 t^{-5}+4 t+C \end{aligned}

$b)$ \begin{aligned}
\int\left(x^8+x^{-8}\right) d x & =\int x^8 d x+\int x^{-8} d x \\
& =\frac{x^9}{9}+\frac{x^{-7}}{-7} \\
& =\frac{x^9}{9}-\frac{x^{-7}}{7}
\end{aligned}

$c)$ \begin{aligned} \int\left(3 \sqrt[4]{x^3}+\frac{7}{x^5}+\frac{1}{6 \sqrt{x}}\right) d x \\
&=3 \int \sqrt[4]{x^3} d x+7 \int x^{-5} d x+\frac{1}{6} \int x^{-\frac{1}{2}} d x\\
&=3 \int x^{\frac{3}{4}} d x+7 \int x^{-5} d x+\frac{1}{6} \int x^{-\frac{1}{2}} d x\\
&=\frac{4.3}{7} x^{\frac{7}{4}}-\frac{7}{4} x^{-4}+\frac{1}{3} x^{\frac{1}{2}}+C \\
&= \frac{12}{7} x^{\frac{7}{4}}-\frac{7}{4} x^{-4}+\frac{x^{\frac{1}{2}}}{3}+C \end{aligned}.

$d)$ \begin{aligned}
\int d y & =\int 1 d y \\
& =y+C
\end{aligned}

$e)$ Commençons par développer l’expression $(\omega+\sqrt[3]{\omega})\left(4-\omega^2\right)$ :

\begin{aligned}
(w+\sqrt[3]{\omega})\left(4-w^2\right) & =4 w-w^3+4 \sqrt[3]{w}-w^2 \cdot \sqrt[3]{\omega} \\
& =4 w-w^3+4 w^{\frac{1}{3}}-w^2 \cdot w^{\frac{1}{3}} \\
& =4 w-w^3+4 w^{\frac{1}{3}}-w^{\frac{6}{3}+\frac{1}{3}} \\
& =4 w-w^3+4 w^{1 / 3}-w^{7 / 3}
\end{aligned}

Ainsi \begin{aligned}
\int(\omega+\sqrt[3]{\omega})\left(4-\omega^2\right) d \omega & =\int\left(4 \omega-\omega^3+4 w^{1 / 3}-\omega^{7 / 3}\right) d \omega \\
& =4 \int \omega d \omega-\int \omega^3 d \omega+4 \int \omega^{1 / 3} d \omega-\int \omega^{7 / 3} d \omega \\
& =\frac{4 \omega^2}{2}-\frac{\omega^4}{4}+3 \omega^{4 / 3}-\frac{3}{10} w^{10 / 3} \\
& =2 \omega^2-\frac{\omega^4}{4}+3 \omega^{4 / 3}-\frac{3}{10} w^{10 / 3}
\end{aligned}