Respuesta :
[tex]\bf \displaystyle \int\limits_{0}^{7}\ 9x\sqrt[3]{3x+1}\cdot dx\\\\
-----------------------------\\\\
u=3x+1\implies \cfrac{du}{dx}=3\implies \cfrac{du}{3}=dx\\\\
-----------------------------\\\\
\displaystyle \int\limits_{0}^{7}\ 9x\sqrt[3]{u}\cdot \cfrac{du}{3}\implies \int_{0}^{7}\ 3x\sqrt[3]{u}\cdot dx\\\\
-----------------------------\\\\
now\qquad u=3x+1\implies u-1=3x\\\\
-----------------------------\\\\[/tex]
[tex]\bf \displaystyle \int\limits_{0}^{7}\ (u-1)u^{\frac{1}{3}}\cdot dx\implies \int\limits_{0}^{7}\ \left( u^{\frac{4}{3}}-u^{\frac{1}{3}} \right) dx\\\\ -----------------------------\\\\ \textit{now, let's change the bounds, using u(x)} \\\\\\ u(x)=3x+1\qquad thus\qquad u(0)=1\qquad u(7)=22\\\\ -----------------------------\\\\[/tex]
[tex]\bf \displaystyle \int\limits_{1}^{22}\ \left( u^{\frac{4}{3}}-u^{\frac{1}{3}} \right) dx\implies \cfrac{u^{\frac{7}{3}}}{\frac{7}{3}}-\cfrac{u^{\frac{4}{3}}}{\frac{4}{3}}\implies \left. \cfrac{3\sqrt[3]{u^7}}{7}-\cfrac{3\sqrt[3]{u^4}}{4}\right]_{1}^{22}[/tex]
upper-bound part
[tex]\bf \left[ \cfrac{3\cdot 1356.187}{7} \right]-\left[ \cfrac{3\cdot 61.645}{4} \right] \\\\\\ 581.22-46.234\approx 534.98936648870[/tex]
and lower-bound part
[tex]\bf \left[ \cfrac{3}{7} \right]-\left[ \cfrac{3}{4} \right]\implies -\cfrac{9}{28} \\\\\\ thus \\\\\\ 534.98936648870355525281-\left( -\cfrac{9}{28} \right) \\\\\\ 534.98936648870355525281+\cfrac{9}{28} \approx 535.31079506013212668138[/tex]
[tex]\bf \displaystyle \int\limits_{0}^{7}\ (u-1)u^{\frac{1}{3}}\cdot dx\implies \int\limits_{0}^{7}\ \left( u^{\frac{4}{3}}-u^{\frac{1}{3}} \right) dx\\\\ -----------------------------\\\\ \textit{now, let's change the bounds, using u(x)} \\\\\\ u(x)=3x+1\qquad thus\qquad u(0)=1\qquad u(7)=22\\\\ -----------------------------\\\\[/tex]
[tex]\bf \displaystyle \int\limits_{1}^{22}\ \left( u^{\frac{4}{3}}-u^{\frac{1}{3}} \right) dx\implies \cfrac{u^{\frac{7}{3}}}{\frac{7}{3}}-\cfrac{u^{\frac{4}{3}}}{\frac{4}{3}}\implies \left. \cfrac{3\sqrt[3]{u^7}}{7}-\cfrac{3\sqrt[3]{u^4}}{4}\right]_{1}^{22}[/tex]
upper-bound part
[tex]\bf \left[ \cfrac{3\cdot 1356.187}{7} \right]-\left[ \cfrac{3\cdot 61.645}{4} \right] \\\\\\ 581.22-46.234\approx 534.98936648870[/tex]
and lower-bound part
[tex]\bf \left[ \cfrac{3}{7} \right]-\left[ \cfrac{3}{4} \right]\implies -\cfrac{9}{28} \\\\\\ thus \\\\\\ 534.98936648870355525281-\left( -\cfrac{9}{28} \right) \\\\\\ 534.98936648870355525281+\cfrac{9}{28} \approx 535.31079506013212668138[/tex]
