Mathematics - Indefinite Integration Question with Solution | TestHub

$\begin{array}{rl}& \text{ Let }3-x^2=t^2\\ & +xdx=-tdt\\ & f(x)=\int \left(3-t^2\right)\cdot t(-tdt)+c\\ & =\int \left({\mathrm{t}}^4-3{\mathrm{t}}^2\right)\mathrm{dt}+\mathrm{c}\\ & =\frac{{\mathrm{t}}^5}{5}-{\mathrm{t}}^3+\mathrm{c}\\ & \mathrm{f}(\mathrm{x})=\frac{{\left(3-{\mathrm{x}}^2\right)}^{5/2}}{5}-{\left(3-{\mathrm{x}}^2\right)}^{3/2}+\mathrm{c}\\ & \mathrm{f}(\sqrt{2})=\frac{1}{5}-1+\mathrm{c}=-\frac{4}{5}\end{array}$
$\begin{array}{rl}& \mathrm{c}=0\\ & \mathrm{f}(1)=\frac{2^{5/2}}{5}-2^{3/2}\\ & =2^{1/2}\left(\frac{4}{5}-2\right)\\ & \mathrm{f}(1)-\frac{-6\sqrt{2}}{5}\end{array}$