Question

Ensino superior calculo II
Calcule o valor da integral usando o método da substituição trigonométrica

Answer 1

$\large\boxed{\begin{array}{l}\rm Quando~o~integrando~tem~a~forma\\\rm\sqrt{a^2+x^2},use~a~substituic_{\!\!,}\tilde ao~x=a~tg(\theta)\\\rm~onde~dx=a~sec^2(\theta)~d\theta\\\rm e~o~radicando~\sqrt{a^2+x^2}~se~torna~\bf a\,sec(\theta).\end{array}}$

$\boxed{\begin{array}{l}\underline{\rm observe~a~figura~que~anexei.}\\\sf vamos~usar~a~substituic_{\!\!,}\tilde ao~x=7tg(\theta)\\\sf ent\tilde ao~dx=7sec^2(\theta)~d\theta~e~\sqrt{x^2+49}=7sec(\theta).\\\rm assim:\\\displaystyle\sf\int\dfrac{1}{\sqrt{x^2+49}}~dx=\int\dfrac{\diagup\!\!\!7~\backslash\!\!\!\!\!sec^2(\theta)}{\diagup\!\!\!7~~~\backslash\!\!\!\!\!sec(\theta)}~d\theta\\\displaystyle\sf\int sec(\theta)~d\theta=\ell n|sec(\theta)+tg(\theta)|+C\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm usando~o~tri\hat angulo~auxiliar~temos\!:} \\\sf tg(\theta)=\dfrac{cat~oposto}{cat~adj}=\dfrac{x}{7}\\\sf sec(\theta)=\dfrac{hip}{cat~adj}=\dfrac{\sqrt{x^2+49}}{7}\\\underline{\rm substituindo~temos:}\\\displaystyle\sf\int\dfrac{1}{\sqrt{x^2+49}}~dx=\ell n\bigg|\dfrac{x+\sqrt{x^2+49}}{7}\bigg|+C\end{array}}$