[Integrales Indefinidas] Integrar, desarrollamos la función a integrar

Integrar $\int\dfrac{(x^m+x^n)^2}{\sqrt{x}}dx$.

Solución:

Desarrollamos la función a integrar
$$\begin{align}\dfrac{(x^m+x^n)^2}{\sqrt{x}}&=\dfrac{x^{2m}+2x^{m+n}+x^{2n}}{x^{\frac{1}{2}}}\\ &=x^{2m-\frac{1}{2}}+2x^{m+n-\frac{1}{2}}+x^{2n-\frac{1}{2}}\\ &=x^{\frac{4m-1}{2}}+2x^{\frac{2m+2n-1}{2}}+x^{\frac{4n-1}{2}}\end{align}$$
Integramos
$$\begin{align}\int\dfrac{(x^m+x^n)^2}{\sqrt{x}}dx&=\int\left(x^{\frac{4m-1}{2}}+2x^{\frac{2m+2n-1}{2}}+x^{\frac{4n-1}{2}}\right)dx\\ &=\dfrac{x^{\frac{4m+1}{2}}}{\dfrac{4m+1}{2}}+\dfrac{2x^{\frac{2m+2n+1}{2}}}{\dfrac{2m+2n+1}{2}}+\dfrac{x^{\frac{4n+1}{2}}}{\dfrac{4n+1}{2}}+c\\ &=\dfrac{2x^{\frac{4m+1}{2}}}{4m+1}+\dfrac{4x^{\frac{2m+2n+1}{2}}}{2m+2n+1}+\dfrac{2x^{\frac{4n+1}{2}}}{4n+1}+c\end{align}$$