Theorem 6.1.2 (Differentiation of the Integral) If f is continuous on [a, b), then F(x) = S f(t) dt is C' and F' = f. Theorem 6.1.3 (Integration of the Derivative) If f is C' on [a, b), then S f'(x) dx = f(b) – f(a).
Theorem 6.1.2 (Differentiation of the Integral) If f is continuous on [a, b), then F(x) = S f(t) dt is C' and F' = f. Theorem 6.1.3 (Integration of the Derivative) If f is C' on [a, b), then S f'(x) dx = f(b) – f(a).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove Theorem 6.1.3 implies Theorem 6.1.2
![**Theorem 6.1.2 (Differentiation of the Integral)**: If \( f \) is continuous on \([a, b]\), then \( F(x) = \int_{a}^{x} f(t) \, dt \) is \( C^1 \) and \( F' = f \).
**Theorem 6.1.3 (Integration of the Derivative)**: If \( f \) is \( C^1 \) on \([a, b]\), then \(\int_{a}^{b} f'(x) \, dx = f(b) - f(a)\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbd1f10ad-6c07-45d3-868c-40133678a6db%2Fc0875c5e-d0e6-4fca-a224-f0b821335456%2Fvbyn3a9o_processed.png&w=3840&q=75)
Transcribed Image Text:**Theorem 6.1.2 (Differentiation of the Integral)**: If \( f \) is continuous on \([a, b]\), then \( F(x) = \int_{a}^{x} f(t) \, dt \) is \( C^1 \) and \( F' = f \).
**Theorem 6.1.3 (Integration of the Derivative)**: If \( f \) is \( C^1 \) on \([a, b]\), then \(\int_{a}^{b} f'(x) \, dx = f(b) - f(a)\).
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