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Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int 4 x^{3} \sqrt{5-x^{2}} d x$

Solution:

Assume $5-x^{2}=t^{2}$

$d\left(5-x^{2}\right)=d\left(t^{2}\right)$

$-2 x \cdot d x=2 t \cdot d t$

$\Rightarrow x d x=-t . d x$

$\Rightarrow \mathrm{dx}=\frac{-\mathrm{t}}{\mathrm{x}} \mathrm{dt}$

Substituting $t$ and $d t$

$\Rightarrow \int 4 x^{3} \sqrt{t^{2}} \frac{-t}{x} d t$

$\Rightarrow 4 \int x^{2} t^{2}$

$\Rightarrow x^{2}=5-t^{2}$

$\Rightarrow 4 \int\left(5-t^{2}\right) t^{2} \cdot d t$

$\Rightarrow 20 \int \mathrm{t}^{2} \mathrm{dt}-4 \int \mathrm{t}^{4} \mathrm{dt}$

$\Rightarrow 20 \times \frac{t^{3}}{3}-4 \frac{t^{5}}{5}+c$

$\Rightarrow 20\left(5-x^{2}\right)^{3 \backslash 2}-\frac{4}{5}\left(5-x^{2}\right)^{5 \backslash 2}+c$

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