TMUA 2020 · Paper 2 · Question 16

The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial f :

\frac{d}{d x} (\int_{0}^{x} f (t) d t) = f (x)

A student calculates $\frac{d}{d x} (\int_{x}^{2 x} t^{2} d t)$ as follows:

(I) $\int_{x}^{2 x} t^{2} d t = \int_{0}^{2 x} t^{2} d t - \int_{0}^{x} t^{2} d t$ (II) By FTC, $\frac{d}{d x} (\int_{0}^{x} t^{2} d t) = x^{2}$ (III) By FTC, $\frac{d}{d x} (\int_{0}^{2 x} t^{2} d t) = (2 x)^{2} = 4 x^{2}$ (IV) So $\frac{d}{d x} (\int_{x}^{2 x} t^{2} d t) = 4 x^{2} - x^{2}$ (V) giving $\frac{d}{d x} (\int_{x}^{2 x} t^{2} d t) = 3 x^{2}$

Which of the following best describes the student's calculation?