f(x)= x^3√3-x^2dx5f(√2)=-4f(1)

Question

$f(x)=\int {x}^{3}\sqrt{3-{x}^{2}}dx5f(\sqrt{2})=-4f(1)$

Accepted Answer

$\begin{gathered} \int {x}^{3}\sqrt{3-{x}^{2}}dx \ 3-{x}^{2}={t}^{2} \ -2xdx=2tdt \ =-\int {t}^{2}(3-{t}^{2})dt=\int {t}^{4}-3{t}^{2}dt \ =\frac{{t}^{5}}{5}-{t}^{3}+c \end{gathered}$

$\begin{gathered} f(x)=\frac{{(3-{x}^{2})}^{\frac{5}{2}}}{5}-{(3-{x}^{2})}^{\frac{3}{2}}+c \ \because f(\sqrt{2})=-\frac{4}{5} \ \Rightarrow \, -\frac{4}{5}=\frac{1}{5}-1+c\Rightarrow c=0 \ 	herefore \, f(x) =\frac{{(3-{x}^{2})}^{\frac{5}{2}}}{5}-{(3-{x}^{2})}^{\frac{3}{2}} \ f(1) =\frac{{2}^{\frac{5}{2}}}{5}-{2}^{\frac{3}{2}}={2}^{\frac{3}{2}}[\frac{2}{5}-1] \ =2\sqrt{2}(-\frac{3}{5})=-\frac{6}{5}\sqrt{2} \end{gathered}$

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