Question

Calcule a integral por substituição trigonométrica
HELP????

Answer 1

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

$\boxed{\begin{array}{l}\rm vamos~usar~a~substituic_{\!\!,}\tilde ao~x=\dfrac{3}{5}~tg(\theta)\\\sf de~forma~que~dx=\dfrac{3}{5}~sec^2(\theta)\\\sf e~o~radicando~\sqrt{x^2+\dfrac{9}{25}}~se~torna~\dfrac{3}{5}sec(\theta)\\\underline{\rm substituindo~temos\!:}\\\displaystyle\sf\int\dfrac{dx}{\sqrt{x^2+\frac{9}{25}}}=\int\dfrac{\diagup\!\!\!\!\frac{3}{5}~\backslash\!\!\!\!sec^2(\theta)}{\diagup\!\!\!\!\frac{3}{5}~\backslash\!\!\!\!sec(\theta)}~d\theta\end{array}}$

$\boxed{\begin{array}{l}\displaystyle\sf=\int sec(\theta)~d\theta=\ell n|sec(\theta)+tg(\theta)|+C\\\underline{\rm usando~o~tri\hat angulo~auxiliar~temos\!:}\\\sf tg(\theta)=\dfrac{cat~op}{cat~adj}=\dfrac{x}{\frac{3}{5}}=\dfrac{5x}{3}\\\sf sec(\theta)=\dfrac{hip}{cat~adj}=\dfrac{\sqrt{x^2+\frac{9}{25}}}{\frac{3}{5}}=\dfrac{5\sqrt{x^2+\frac{9}{25}}}{3}\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm substituindo~temos\!:}\\\displaystyle\sf\int\dfrac{dx}{\sqrt{x^2+\frac{9}{25}}}=\ell n\bigg|\dfrac{5(x+\sqrt{x^2+\frac{9}{25}})}{3}\bigg|+C\end{array}}$