Dúvida 1 / X SQRT (X^2 +1)

por **Knoner** » Qui Set 26, 2013 20:15

Olá, estou em dúvida na resolução da seguinte integral: 1 / X SQRT (X^2 +1)

Obrigado !

por **Man Utd** » Qui Set 26, 2013 23:02

olá

$\\\\\\ \int \frac {1}{x*\sqrt{x^{2}+1}}dx \\\\ x=tg\beta \Leftrightarrow dx=sec^{2}\beta d\beta \\\\\\ \int \frac {sec^{2}\beta }{tg\beta*\sqrt{(tg\beta)^{2}+1}}d\beta \\\\\\ \int \frac {sec^{2}\beta }{tg\beta*sec\beta}d\beta \\\\\\ \int \frac {sec\beta }{tg\beta}d\beta \\\\\\ \int \frac{\frac{1}{cos\beta}}{\frac{sen\beta}{cos\beta}}d\beta \\\\\\ \int cossec \beta d\beta$

vamos aplicar uma técnica para integrar $\int cossec \beta d\beta$ :

$\\\\\\ \int \frac{(cossec \beta)*(cotg\beta+cossec\beta)}{(cotg\beta+cossec\beta)} d\beta \\\\\\ -\int \frac{-cotg\beta*cossec\beta-cossec^{2}\beta}{(cotg\beta+cossec\beta)} \\\\\\ s=cotg\beta+cossec\beta \Leftrightarrow ds= -cotg\beta*cossec\beta-cossec^{2}\beta d\beta \\\\\\ - \int \frac{1}{s}ds \\\\ -ln|s|+C \\\\ -ln|cotg\beta+cossec\beta |+C \\\\ -ln|\frac{1}{x}+\frac{\sqrt{x^{2}+1}}{x}|+C \\\\ -ln|\frac{\sqrt{x^{2}+1}+1}{x}|+C \\\\ -(ln|\sqrt{x^{2}+1}+1|-ln|x|)+C \\\\ -ln|\sqrt{x^{2}+1}+1|+ln|x|+C \\\\ ln|\frac{x}{\sqrt{x^{2}+1}+1}|+C$

edit:resposta editada