Ejercicios Integrales

Ejercicios_Resueltos integrales integrales inmediatas

Resuelve la integral:

– $\int \frac{2+x^2}{\sqrt{x}} \: dx$

SOLUCIÓN

Linealidad:

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

$= \displaystyle\int \dfrac{2}{\sqrt{x}}\,dx + \displaystyle\int \dfrac{x^{2}}{\sqrt{x}}\,dx$

Potencia:

$\displaystyle\int \dfrac{2}{\sqrt{x}}\,dx = \displaystyle\int 2x^{-1/2}\,dx = 2\displaystyle\int x^{-1/2}\,dx = 2\cdot\dfrac{x^{-1/2+1}}{-1/2+1} = 2\cdot\dfrac{x^{1/2}}{1/2} = 4\sqrt{x}$

$\displaystyle\int \dfrac{x^{2}}{\sqrt{x}}\,dx = \displaystyle\int x^{3/2}\,dx = \dfrac{x^{3/2+1}}{3/2+1} = \dfrac{x^{5/2}}{5/2} = \dfrac{2}{5}x^{5/2}$

$= 4\sqrt{x} + \dfrac{2}{5}x^{5/2} + C$

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