Chapter 4 - Integration - 4.6 The Fundamental Theorem Of Calculus - Exercises Set 4.6 - Page 321: 51

See explanation.

Recall that: \[ \int_{a}^{b} f(x) d x=-\int_{b}^{a} f(x) d x \] And therefore \[ \frac{d}{d x} \int_{x}^{0} t \sec t d t=-\frac{d}{d x} \int_{0}^{x} t \sec t d t=-x \sec x \] Remember the Fundamental Theorem of Calculus: \[ \frac{d}{d x} \int_{a}^{x} f(t) d t=f(x) \]