int (sqrt (28+343x^2m))/(x^1-m)dx

Para resolver esta integral, primero vamos a hacer un cambio de variable. Sea \( u = 7x^{m} \), entonces \( du = 7mx^{m-1}dx \) o \( dx = \frac{du}{7mx^{m-1}} \).<br /><br />Sustituyendo \( u \) y \( dx \) en la integral, obtenemos:<br /><br />\[ \int \frac{\sqrt{28 + 343x^{2m}}}{x^{1-m}}dx = \int \frac{\sqrt{28 + 343u^2}}{x^{1-m}} \cdot \frac{du}{7mx^{m-1}} \]<br /><br />Simplificando, obtenemos:<br /><br />\[ \int \frac{\sqrt{28 + 343u^2}}{x^{1-m}} \cdot \frac{du}{7mx^{m-1}} = \frac{1}{7m} \int \frac{\sqrt{28 + 343u^2}}{u} du \]<br /><br />Ahora, podemos resolver esta integral utilizando una sustitución adicional. Sea \( v = \sqrt{28 + 343u^2} \), entonces \( dv = \frac{343u}{\sqrt{28 + 343u^2}} du \) o \( du = \frac{dv}{343u/\sqrt{28 + 343u^2}} \).<br /><br />Sustituyendo \( v \) y \( du \) en la integral, obtenemos:<br /><br />\[ \frac{1}{7m} \int \frac{\sqrt{28 + 343u^2}}{u} du = \frac{1}{7m} \int \frac{v}{u} \cdot \frac{dv}{343u/\sqrt{28 + 343u^2}} \]<br /><br />Simplificando, obtenemos:<br /><br />\[ \frac{1}{7m} \int \frac{v}{u} \cdot \frac{dv}{343u/\sqrt{28 + 343u^2}} = \frac{1}{7m} \cdot \frac{1}{343} \int \frac{v}{u} \cdot \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{v}{u} \cdot \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{v}{u} \cdot \frac{dv}{v} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{u} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ = \frac{1}{7m \cdot 343} \int \frac{dv}{\sqrt{28 + 343u^2}} \]<br /><br />\[ =