➡️ Sea $sen^m(x)$, con $m= 2k+1$ impar, entonces:

Se descompone $\quad$➡︎$\quad$ $sen^{2k+1}(x)=sen^{2k}(x).sen(x)$
Se reorganiza $\quad$➡︎$\quad$ $sen^{2k}(x).sen(x)=(sen^2(x))^k.sen(x)$
Se utiliza la identidad $\space$ $sen^2(x)$ $\quad$➡︎$\quad$ $(1-cos^2(x))^k.sen(x)$

$$ \int (1-cos^2(x))^k(x).sen(x) \, dx = -\int (1-u^2)^k du$$ donde, $u=cos(x)$ y $du= -sen(x)dx$.

➡️ Sea $cos^m(x)$, con $m= 2k+1$ impar, entonces:

Se descompone $\quad$➡︎$\quad$ $cos^{2k+1}(x)=cos^{2k}(x).cos(x)$
Se reorganiza $\quad$➡︎$\quad$ $cos^{2k}(x).cos(x)=(cos^2(x))^k.cos(x)$
Se utiliza la identidad $\space$ $cos^2(x)$ $\quad$➡︎$\quad$ $(1-sen^2(x))^k.cos(x)$

$$ \int (1-sen^2(x))^k(x).cos(x) \, dx = -\int (1-u^2)^k du$$ donde, $u=sen(x)$ y $du= cos(x)dx$.