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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) §5.6 Properties of the Integral and Average Value, #22 · Calculus · Integrals

(a) If … is increasing, explain from Property 6 why … for…

(a) If $v (x)$ is increasing, explain from Property 6 why $\int_{0}^{x} v (t) d t \leq xv (x)$ for $x > 0$ .
(b) Take derivatives of both sides for a second proof.

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Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §5.6 Properties of the Integral and Average Value, #22

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