Question

Me ajuda a calcular a integral PORFAVOOR ♥
$\int\limits\frac{ {2 \sqrt{x}+ x} }{4x} \, dx$

Answer 1

Boa noite Marly!

Solução! 


$\displaystyle \int \frac{2 \sqrt{x} +x}{4x}dx$


Primeiro vamos reescrever a  integral!


$\displaystyle \int \frac{2 \sqrt{x} }{4x}+ \frac{x}{4x}dx\\\\\\\\\\\ \dfrac{1}{4} \displaystyle \int \frac{2 \sqrt{x} }{x}+ \dfrac{x}{x}dx\\\\\\\\\ \dfrac{1}{4} \displaystyle \int \frac{2 \sqrt{x} }{x}+ 1dx$


$\dfrac{1}{4} \displaystyle \int 2x^{ \frac{1}{2} }.x^{-1} + 1dx\\\\\\\\\ \dfrac{1}{4} \displaystyle \int2 x^{ -\frac{1}{2} }) + 1dx\\\\\\\\\ I= \dfrac{1}{4}( \frac{ 2x^{ -\frac{1}{2} +1}}{ -\frac{1}{2} +1} )+1\\\\\\\\ I= \dfrac{1}{4}( \frac{ 2x^{ \frac{1}{2}}}{ \frac{1}{2} } )+x\\\\\\\\ I= \dfrac{1}{4}( 4 \sqrt{x} )+x\\\\\\\\ I= \frac{2 \sqrt{x} +x}{4}+c$


$\boxed{Resposta:\displaystyle \int \frac{2 \sqrt{x} +x}{4x}dx=\frac{2 \sqrt{x} +x}{4}+c}$


Boa noite!
Bons estudos!

Answer 2

Oi Marlyn 

∫ (2√x + x)/(4x) dx = 1/2 ∫ 1/√x + ∫ 1/4 dx

∫ 1/√x = 2√x 

∫ (2√x + x)/(4x) dx = √x + x/4 + C

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