Problem

2) $\int y^{3} \sqrt{3-y^{2}} d y$

Answer

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Answer

Calculate the integral in terms of $u$ and substitute back $y$ to get the final answer: \(\boxed{\frac{9\sqrt{3}}{5}(1 - y^2)^2 - \frac{3\sqrt{3}}{5}(1 - y^2)y^2 - \frac{6\sqrt{3}}{5}(1 - y^2)}\)

Steps

Step 1 ：Substitute $y = \sqrt{3} \sin{u}$, then $dy = \sqrt{3} \cos{u} du$

Step 2 ：Calculate the integral in terms of $u$ and substitute back $y$ to get the final answer: \(\boxed{\frac{9\sqrt{3}}{5}(1 - y^2)^2 - \frac{3\sqrt{3}}{5}(1 - y^2)y^2 - \frac{6\sqrt{3}}{5}(1 - y^2)}\)

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