#title:Regra de integra\c{c}\~{a}o P $\frac{u'}{\sqrt{1+u^2}}=arcsinh(u)+C$
#Let:



f u= [x, sin(x), cos(x), exp(x) ]; 
f du=[1, cos(x), -sin(x),exp(x) ];
f raiz=[sqrt(1+x^2), sqrt(1+(sin(x))^2),sqrt(1+(cos(x))^2),sqrt(1+(exp(x))^2)];

u~du~raiz;



cf res = arcsinh(#u);



#Question:

\noindent Calcule o integral indefinido,
$$
\int (#du)/(#raiz) dx.
$$


#Sugestion:

\noindent Utilize a f\'{o}rmula do formul\'{a}rio
$$
\int \frac{u'}{\sqrt{1+u^2}} dx= arcsinh(u)+C.
$$



#Resolution

Obtem-se
$$
\int (#du)/(#raiz) dx = #res+C.
$$

#result

$$ #res+C $$

#Verify

arcsinh(#u).