Let $f(t)=\ln (t)$. Then, $\frac{d}{d x}\left(\int\limits_{x^2}^{x^3} f(t) d t\right)$.

Which one of the above is incorrect?

has value 0 when $x=0$

Let

$F(x)=\frac{d}{d x}\left(\int\limits_{x^2}^{x^3} \ln (t) d t\right)$

$\Rightarrow F'(x) =3 x^2 \log x^3-2 x \log x^2$       [Using Leibnitz's Rule]

$\Rightarrow F'(x) =9 x^2 \log x-4 x \log x=\left(9 x^2-4 x\right) \log x$

∴  $F'(x)=(9 x-4)+(18 x-4) \log x$

Clearly, F(x) and F'(x) satisfy (b), (c) and (d) respectively.

Hence, option (a) is incorrect.