Question

Desarrollar la siguiente integral definida

Answer 1

$\displaystyle{\int_{\frac{\pi}{2}}^{\pi}\left[\sin(x)+\frac{1}{x}\right]dx}$

por propiedad:

$\displaystyle{\int_{\frac{\pi}{2}}^{\pi}\sin(x)dx+\int_{\frac{\pi}{2}}^{\pi}\frac{1}{x}dx}$

$\mathrm{\large{Por \ Tabla:}}$

$\begin{cases}\int \sin(x)dx=-cos(x)+k\cr \int \frac{1}{x}dx=ln(x)+k\end{cases}$

$\mathrm{\large{Aplicando \ barrow:}}$

$\displaystyle{\left[-cos(x)+ln(x)\right]\bigg{|}_{\frac{\pi}{2}}^\pi}$

$\left[-cos(\pi)+ln(\pi)\left]-\left[-cos\left(\frac{\pi}{2}\right)+ln\left(\frac{\pi}{2}\right)\right]$

$\begin{cases}cos(\pi)=-1\cr \cos\left(\frac{\pi}{2}\right)=0\end{cases}$

$-(-1)+ln(\pi)-ln\left(\frac{\pi}{2}\right)$

$\mathrm{Propiedad \ logaritmo:}$

$log_a \left(\frac{b}{c}\right)=log_a (b)-log_a (c)$

$\mathrm{large{Entonces:}}$

$1+ln\left(\frac{\pi}{\frac{\pi}{2}}\right)\Rightarrow$

$1+ln(2)$