Question

integral de x^3/(x^2+3)^3/2

Answer 1

En este caso debemos eliminar primero el exponente 1/2 del denominador. Para ello nos apoyaremos en un cambio de variable


$t=(x^2+3)^{1/2}\to t^2=x^2+3\to 2t~dt=2x~dx\to \boxed{t~dt=x~dx}\\ \\ \texttt{Por otro lado }x^2=t^2-3\\ \\ \displaystyle \int\dfrac{x^3}{(x^2+3)^{3/2}}dx=\int\dfrac{x^2}{(x^2+3)^{3/2}}\cdot x~dx\\ \\ \\ \int\dfrac{x^3}{(x^2+3)^{3/2}}dx=\int \dfrac{t^2-3}{t^3} \cdot t~dt\\ \\ \\ \int\dfrac{x^3}{(x^2+3)^{3/2}}dx=\int \dfrac{t^2-3}{t^2} ~dt\\ \\ \\ \int\dfrac{x^3}{(x^2+3)^{3/2}}dx=\int 1-\dfrac{3}{t^2} ~dt$


$\displaystyle \int\dfrac{x^3}{(x^2+3)^{3/2}}dx=\int dt-3\int\dfrac{1}{t^2} ~dt\\ \\ \\ \int\dfrac{x^3}{(x^2+3)^{3/2}}dx=t+\dfrac{3}{t}+C\\ \\ \texttt{Devolvemos a la variable: }\\ \\ \\ \boxed{\int\dfrac{x^3}{(x^2+3)^{3/2}}dx=\sqrt{x^2+3}+\dfrac{3}{\sqrt{x^2+3}}+C}$