Exo 102 (22-SMN2U04-C3-EX102)

$a)$ \begin{aligned}
\int \frac{4 x^{10}-2 x^4+15 x^2}{x^3} d x&=\int 4 x^7-2 x+15 x^{-1} d x \\
&=4 \int x^7 d x-2 \int x d x+15 \int \frac{1}{x} d x \\
&=\frac{4 x^8}{8}-\frac{2 x^2}{2}+15 \ln |x|+C \\
&=\frac{x^8}{2}-x^2+15 \ln |x|+C
\end{aligned}

$b)$ \begin{aligned}
\int 3 e^x+5 \cos (x) d x & =3 \int e^x d x+5 \int \cos (x) d x \\
& =3 e^x+5 \sin (x)+C
\end{aligned}

$c)$ \begin{aligned} \int \frac{23}{\sqrt{1-y^2}} d y&=2 \int \frac{1}{\sqrt{1-y^2}} d y \\ &=2 \arcsin (y)+C \end{aligned}

$d)$ \begin{aligned} \int \sin \left(\frac{t}{2}\right) \cos \left(\frac{t}{2}\right) d t& =\frac{1}{2} \int \sin (t) d t\\ &=-\frac{\cos (t)}{2}+C \end{aligned}

Notez que $$\sin(t) = \sin \left(\frac{t}{2}+\frac{t}{2}\right)=\sin \left(\frac{t}{2}\right) \cos \left(\frac{t}{2}\right)+\cos \left(\frac{t}{2}\right) \sin \left(\frac{t}{2}\right)$$