Explanation

(a)
Given equation is $\frac{dy}{dx} = \frac{x\log{x^{2} + x}}{\sin{y + y\cos y}}$
$\Rightarrow (\sin{y + y\cos{y)dy = (x\log{x^{2} + x)dx}}}$
On integrating both sides, we get
$\int_{}^{}{(\sin{y + y\cos{y)dy =}}}\int_{}^{}{(x\log{x^{2} + x)dx}}$
$\Rightarrow \ - \cos{y + y\sin{y + \cos{y = \frac{x^{2}}{2}\log{x^{2} - \int_{}^{}{\frac{x^{2}}{2}\frac{1}{x^{2}}2x\ dx + \int_{}^{}{x\ dx + c}}}}}}$
$\Rightarrow y\sin{y = \frac{x^{2}}{2}2\log{x - \int_{}^{}{x\ dx +}\int_{}^{}{x\ dx + c}}}$
$\Rightarrow y\sin{y = x^{2}\log{x + c}}$