Explanation

(d)
$\int_{0}^{x}{f(t)dt = \int_{x}^{1}{t^{2}\ f(t)dt + \frac{x^{16}}{8} + \frac{x^{6}}{3} + a}}$ …(i)
For $x = 1$
$\int_{0}^{1}{f(t)dt = \frac{1}{8} + \frac{1}{3} + a}$ …(ii)
On differentiating Eq.(i) w.r.t. $x$, we get
$f(x) = - x^{2}f(x) + 2x^{15} + 2x^{5}$
$\Rightarrow f(x) = \frac{2(x^{15} + x^{5})}{1 + x^{2}}$
Now, from Eq. (ii)
$2\int_{0}^{1}{\left( \frac{x^{15} + x^{5}}{1 + x^{2}} \right)dx = \frac{1}{8} + \frac{1}{3} + a\ \ \Rightarrow \ a = - \frac{167}{840}}$