Prove-that-x-0-x-f-s-ds-f-x-

Question Number 183112 by CrispyXYZ last updated on 20/Dec/22

Prove that

\frac{\partial}{\partial x} \underset{0}{\int^{x}} f (s) d s = f (x)

Answered by Emrice last updated on 20/Dec/22

Prove that

\frac{\partial}{\partial x} \underset{0}{\int^{x}} f (s) d s = f (x)

o n s a i t q u e

\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x

\Rightarrow \int_{0}^{x} f (s) d s = \int_{0}^{x} f (0 + x - s) d s

\Rightarrow \frac{\partial}{\partial x} \int_{0}^{x} f (s) d s = \int_{0}^{x} \frac{\partial f (x - s)}{\partial x} d s

= \int_{0}^{x} f^{'} (x - s) d s

= \int_{0}^{x} f^{'} (s) d s

= f (x)

Answered by mr W last updated on 21/Dec/22

s a y \frac{d F (x)}{d x} = f (x)

t h e n \int_{0}^{x} f (s) d s = F (x) - F (0)

\frac{d}{d x} \int_{0}^{x} f (s) d s = \frac{d F (x)}{d x} - \frac{d F (0)}{d x} = f (x) ✓

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