Question

calcular a integral utilizando o método da substituição trigonométrica

Answer 1

$\Large\boxed{\begin{array}{l}\underline{\rm Integrac_{\!\!,}\tilde ao~por~substituic_{\!\!,}\tilde ao}\\\underline{\rm trigonom\acute etrica}\\\sf Quando~o~integrando~\acute e~da~forma~\sqrt{a^2+x^2}\\\sf use~a~substituic_{\!\!,}\tilde ao~x=a\,tg(\theta)~de~modo~que\\\sf a~express\tilde ao~\sqrt{a^2+x^2}~se~torna~a\,sec(\theta)\\\sf e~dx~se~torne~a\,sec^2(\theta)d\theta\end{array}}$

$\Large\boxed{\begin{array}{l}\displaystyle\sf\int\dfrac{dx}{\sqrt{\frac{49}{36}+x^2}}\\\sf fac_{\!\!,}a~x=\dfrac{7}{6}\,tg(\theta)\\\sf de~modo~que~\sqrt{\dfrac{49}{36}+x^2}=\dfrac{7}{6}sec(\theta)\\\sf e~dx=\dfrac{7}{6}\,sec^2(\theta)d\theta\end{array}}$

$\boxed{\begin{array}{l}\displaystyle\sf\int\dfrac{dx}{\sqrt{\dfrac{49}{36}+x^2}}=\int\dfrac{\diagup\!\!\!\!\!\!\frac{7}{6}~\diagup\!\!\!\!\!\!sec^2(\theta)d\theta}{\diagup\!\!\!\!\!\!\frac{7}{6}~\diagup\!\!\!\!\!sec(\theta)}\\\displaystyle\sf=\int sec(\theta)d\theta=\ell n|sec(\theta)+tg(\theta)|+k\\\underline{\rm usando~o~tri\hat angulo~auxiliar~temos:}\end{array}}$

$\Large\boxed{\begin{array}{l}\sf sec(\theta)=\dfrac{\sqrt{\frac{49}{36}+x^2}}{\frac{7}{6}}=\dfrac{6\sqrt{\frac{49}{36}+x^2}}{7}\\\sf e~~tg(\theta)=\dfrac{x}{\frac{7}{6}}=\dfrac{6x}{7}\\\underline{\rm substituindo~temos:}\end{array}}$

$\Large\boxed{\begin{array}{l}\displaystyle\sf\int\dfrac{dx}{\sqrt{\frac{49}{36}}+x^2}=\ell n\bigg|\dfrac{6\cdot\bigg(\sqrt{\frac{49}{46}+x^2}+x\bigg)}{7}\bigg|+k\end{array}}$

$\Large\boxed{\begin{array}{l}\displaystyle\sf\ell ife=\int_{birth}^{death}\dfrac{happiness}{time}d_{time}\end{array}}$