Question
Apply Bonnet Mean Value Theorem for integrals to show that
Answer :
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Consider the integral [ \int_7^{10} \frac{\sin x}{x} , dx. ] Let (f(x) = \sin x) and (g(x) = \frac{1}{x}). Both functions are continuous on ([7,10]) and (g(x)) is positive and monotonic decreasing. Bonnet’s Mean Value Theorem for integrals states that if (f) is continuous and (g) is monotonic, then there exists (\xi \in [7,10]) such that [ \int_7^{10} f(x) g(x) , dx = f(7) \int_7^{\xi} g(x) , dx + f(10) \int_{\xi}^{10} g(x) , dx ] or equivalently, the integral is bounded by [ \min f(x) \int_7^{10} g(x) , dx \le \int_7^{10} f(x) g(x) , dx \le \max f(x) \int_7^{10} g(x) , dx. ]
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