Question 1

Evaluate the following intergration:
∫y³ 3√(y.dy)

Question 2

Evaluate:

$\: \: \: \: \: \: \displaystyle\int\rm {y}^(2) \sqrt[3]{y} \: dy$

${\large\underline{\sf{Solution-}}}$

$\: \: \: \: \: \: \displaystyle\int\rm {y}^(2) \sqrt[3]{y} \: dy$

can be rewritten as

$\rm \:  =  \: \displaystyle\int\rm {y}^(2) * {\bigg(y\bigg) }^{(1)/(3) } \: dy$

$\rm \:  =  \: \displaystyle\int\rm {\bigg(y\bigg) }^{2 + (1)/(3) } \: dy$

$\rm \:  =  \: \displaystyle\int\rm {\bigg(y\bigg) }^{(6 + 1)/(3) } \: dy$

$\rm \:  =  \: \displaystyle\int\rm {\bigg(y\bigg) }^{(7)/(3) } \: dy$

$\rm \:  =  \: \frac{ {\bigg(y\bigg) }^{(7)/(3) + 1} }{(7)/(3) + 1 } + c$

$\rm \:  =  \: \frac{ {\bigg(y\bigg) }^{(7 + 3)/(3)} }{(7 + 3)/(3)} + c$

$\rm \:  =  \: \frac{ {\bigg(y\bigg) }^{(10)/(3)} }{(10)/(3)} + c$

$\rm \:  =  \:(3)/(10) {\bigg(y\bigg) }^{(10)/(3)} + c$

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