Question

Integral indefinida
ex) ∫(√ x+1)( x−√ x+1)dx 
 

Answer 1

$\int\limits { (\sqrt{x} +1)*(x- \sqrt{x} +1)} \, dx$

podemos reescrever a  raíz como 
$\sqrt{x} =x^ \frac{1}{2}$

reescrevendo a equação
$(x^ \frac{1}{2} +1)*(x-x^ \frac{1}{2} +1)$

fazendo a multiplicação

$(x^ \frac{1}{2} +1)*(x-x^ \frac{1}{2} +1)\\\\\\\\(x^ \frac{1}{2}*x)+(x^ \frac{1}{2}*-x^ \frac{1}{2})+(x^ \frac{1}{2}*1)+(1*x)+(1*-x^ \frac{1}{2})+(1*1)$

multiplicação de potencias de mesma base...mante-se a base e soma os expoentes


$(x^ \frac{1}{2}*x)+(x^ \frac{1}{2}*-x^ \frac{1}{2})+(x^ \frac{1}{2}*1)+(1*x)+(1*-x^ \frac{1}{2})+(1*1)\\\\\\(x^ \frac{1}{2}^{+1})+(-x^{ \frac{1}{2}^*{2}})+(x^\frac{1}{2} )+(x)+(-x^ \frac{1}{2} )+(1)\\\\(x^ \frac{3}{2} )+(-x^ \frac{2}{2})+(x^ \frac{1}{2})+(x)+(-x^ \frac{1}{2})+1\\\\(x^ \frac{3}{2} )+(-x})+(x^ \frac{1}{2})+(x)+(-x^ \frac{1}{2})+1=\boxed{x^ \frac{3}{2} +1}$

(cansei kk) ..

agora substituindo na integral

$\int\limits {x^ \frac{3}{2} +1}} \,. dx$

integrando

$x^ \frac{3}{2} +1\\\\ \frac{x^ \frac{3}{2}^{+1} }{\frac{3}{2}^{+1} } +1x\\\\\frac{x^ \frac{5}{2} }{\frac{5}{2} } +x\\\\ \frac{2x^ \frac{5}{2} }{5} +x\\\\ \frac{2 \sqrt{x^5} }{5} +x+k$

K = constante 

$\boxed{\boxed{ \int\limits { (\sqrt{x} +1)*(x- \sqrt{x} +1)} \,. dx }}= \boxed{\boxed{ { \frac{2 \sqrt{x^5} }{5} +x+k} }}$