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If f′f' is continuous on [0,∞)[0,\infty) and lim⁡x→∞f(x)=0\lim_{x \to \infty}f(x)=0, show\newline∫0∞f′(x) dx=−f(0)\int_{0}^{\infty}f'(x)\,dx=-f(0)\newlineWe can extend our definition of average value of a conti function to an infinite interval by defining the average

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Q. If f′f' is continuous on [0,∞)[0,\infty) and lim⁡x→∞f(x)=0\lim_{x \to \infty}f(x)=0, show\newline∫0∞f′(x) dx=−f(0)\int_{0}^{\infty}f'(x)\,dx=-f(0)\newlineWe can extend our definition of average value of a conti function to an infinite interval by defining the average
  1. Understand Given Information: Let's first understand the given information and what we need to prove. We are given that f′(x) f'(x) is continuous on [0,∞) [0, \infty) and that lim⁡x→∞f(x)=0 \lim_{x \to \infty} f(x) = 0 . We need to show that ∫0∞f′(x) dx=−f(0) \int_{0}^{\infty} f'(x) \, dx = -f(0) .
  2. Apply Fundamental Theorem of Calculus: The Fundamental Theorem of Calculus Part 11 states that if F F is an antiderivative of f f on an interval I I , then for any a,b a, b in I I , ∫abf(x) dx=F(b)−F(a) \int_{a}^{b} f(x) \, dx = F(b) - F(a) . In our case, f(x) f(x) is the antiderivative of f′(x) f'(x) , so we can apply this theorem.
  3. Write Integral and Take Limit: We can write the integral of f′(x) f'(x) from 00 to b b (where b b is a finite number that will later approach infinity) as ∫0bf′(x) dx=f(b)−f(0) \int_{0}^{b} f'(x) \, dx = f(b) - f(0) .
  4. Substitute Limit and Simplify: Now, we take the limit of both sides as b b approaches infinity. The left side becomes ∫0∞f′(x) dx \int_{0}^{\infty} f'(x) \, dx , and the right side becomes lim⁡b→∞(f(b)−f(0)) \lim_{b \to \infty} (f(b) - f(0)) .
  5. Finalize the Proof: Since we are given that lim⁡x→∞f(x)=0 \lim_{x \to \infty} f(x) = 0 , we can substitute this into our equation, giving us lim⁡b→∞f(b)=0 \lim_{b \to \infty} f(b) = 0 . Therefore, the right side of our equation becomes 0−f(0) 0 - f(0) , which simplifies to −f(0) -f(0) .
  6. Finalize the Proof: Since we are given that lim⁡x→∞f(x)=0 \lim_{x \to \infty} f(x) = 0 , we can substitute this into our equation, giving us lim⁡b→∞f(b)=0 \lim_{b \to \infty} f(b) = 0 . Therefore, the right side of our equation becomes 0−f(0) 0 - f(0) , which simplifies to −f(0) -f(0) .Putting it all together, we have ∫0∞f′(x) dx=−f(0) \int_{0}^{\infty} f'(x) \, dx = -f(0) , which is what we wanted to prove.

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