[tex]\large \bold {SOLUTION}[/tex]
[tex]\sf{\int\limits^4_2 {x^{3} } \, dx \ > \int\limits^3_2 {x^{2} } \, dx } \\ [/tex]
[tex]\small\textsf{Integrate the polynomials like an indefinite integral}[/tex]
[tex]\small\sf{\int{ {x}^{n} } = \frac{ {x}^{n + 1} }{n + 1 } \,dx} \\[/tex]
[tex]\small\sf{\int{ {x}^{3} } = \frac{ {x}^{3 + 1} }{3 + 1 } \,dx\implies \frac{ {x}^{4} }{4} } \\[/tex]
[tex]\small\sf{\int{ {x}^{2} } = \frac{ {x}^{2 + 1} }{2 + 1 } \,dx\implies \frac{ {x}^{3} }{3} } \\[/tex]
[tex]\small\textsf{Evaluate the limits of integration}[/tex]
[tex]\sf{{\int\limits_{a}^{b} {x}^{n} } \,dx = {b}^{n} - {a}^{n} } \\ [/tex]
[tex]\small\sf{\left. \frac{ {x}^{4} }{4} \: \: \: \right|_{2}^{4} > \left. \frac{ {x}^{3} }{3} \: \: \: \right|_{2}^{3} }\\ [/tex]
[tex]\small\sf{ \dfrac{ {4}^{4} }{4} - \dfrac{ {2}^{4} }{4} } > \dfrac{ {3}^{3} }{3} - \dfrac{ {2}^{3} }{3} [/tex]
[tex]\small\sf{ \dfrac{ 256 }{4} - \dfrac{ 16 }{4} } > \dfrac{ 27 }{3} - \dfrac{ 8 }{3} [/tex]
[tex]\small\sf{ \dfrac{256 - 16}{4} > \dfrac{27 - 8}{3} } [/tex]
[tex]\small\sf{ \dfrac{240}{4} > \dfrac{19}{3} } [/tex]
[tex]\small\sf{60 > 6.33}[/tex]
[tex]\small\sf{\int\limits^4_2 {x^{3} } \, dx \ > \ \int\limits^3_2 {x^{2} } \, dx }\implies \small\boxed{\green{\sf{true }}} \\ [/tex]
[tex]\large\sf{Q.E.D}[/tex]
