Undergrad Why can't we just integrate a simple function?

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The discussion centers on understanding the integration of a function and the concept of the anti-derivative. Participants explore why the integral of a function, specifically ##\ln(5 - 2x)##, involves a multiplier of ##-1/2##. The relationship between differentiation and integration is emphasized, highlighting that the integral is essentially the anti-derivative. Clarification is provided on how differentiating the logarithmic function leads to the appearance of the multiplier. The conversation concludes with a sense of understanding regarding the integration process.
NODARman
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Can anyone explain to me why the second one is the right?
(See the attachment)
PXL_20221016_130317184.jpg
 
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The integral is the anti-derivative. What happens when you differentiate ##\ln(5 - 2x)##?
 
Your question is where does the ##-1/2## multiplier come from?
 
PeroK said:
The integral is the anti-derivative. What happens when you differentiate ##\ln(5 - 2x)##?
Just got it 🙂 👍
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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