Explanation

(d)
Let $I = \int_{0}^{\pi}{\sum_{r = 0}^{3}{a_{r}\cos^{3 - r}{x\sin^{r}{x\ }dx}}}$
$= \int_{0}^{\pi}a_{0}\cos^{3}{x\ dx + \int_{0}^{\pi}{a_{1}\cos^{2}{x\sin{xdx}}}}$
$+ \int_{0}^{\pi}{a_{2}\cos{x\sin^{2}{x\ dx + \int_{0}^{\pi}{a_{3}\sin^{3}{x\ dx}}}}}$
Since, $\int_{0}^{2a}{f(x)dx}$
$= \left\{ \begin{matrix}
2\int_{0}^{a}{f(x)dx\ \ \ \ \ ,if\ \ \ \ f(2a - x) = f(x)} \\
0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ,if\ f(2a - x) = - f(x)
\end{matrix} \right.\$
$\therefore$ Integral Ist and IIIrd become zero.
$\therefore$The given integral is depend upon$\ a_{1}$and$\ a_{3}$.