Evaluate:
[tex] \: \: \: \: \: \: \displaystyle\int\rm {y}^{2} \sqrt[3]{y} \: dy[/tex]
[tex] {\large\underline{\sf{Solution-}}}[/tex]
[tex] \: \: \: \: \: \: \displaystyle\int\rm {y}^{2} \sqrt[3]{y} \: dy[/tex]
can be rewritten as
[tex]\rm \: = \: \displaystyle\int\rm {y}^{2} \times {\bigg(y\bigg) }^{\dfrac{1}{3} } \: dy[/tex]
[tex]\rm \: = \: \displaystyle\int\rm {\bigg(y\bigg) }^{2 + \dfrac{1}{3} } \: dy[/tex]
[tex]\rm \: = \: \displaystyle\int\rm {\bigg(y\bigg) }^{\dfrac{6 + 1}{3} } \: dy[/tex]
[tex]\rm \: = \: \displaystyle\int\rm {\bigg(y\bigg) }^{\dfrac{7}{3} } \: dy[/tex]
[tex]\rm \: = \: \dfrac{ {\bigg(y\bigg) }^{\dfrac{7}{3} + 1} }{\dfrac{7}{3} + 1 } + c[/tex]
[tex]\rm \: = \: \dfrac{ {\bigg(y\bigg) }^{\dfrac{7 + 3}{3}} }{\dfrac{7 + 3}{3}} + c[/tex]
[tex]\rm \: = \: \dfrac{ {\bigg(y\bigg) }^{\dfrac{10}{3}} }{\dfrac{10}{3}} + c[/tex]
[tex]\rm \: = \:\dfrac{3}{10} {\bigg(y\bigg) }^{\dfrac{10}{3}} + c[/tex]
