Undergrad Why is the absolute value of sinx used in the integral of cotx?

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The absolute value of sin(x) is used in the integral of cot(x) to ensure the logarithm remains defined, as ln(sin(x)) is not defined when sin(x) is less than or equal to zero. This requirement is linked to the domains of cot(x) and sin(x). The absolute value ensures that the argument of the logarithm is always positive, allowing for valid integration. The discussion emphasizes the importance of considering the behavior of sin(x) in relation to the logarithmic function. Understanding these domain restrictions is crucial for proper evaluation of the integral.
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Why is there a absolute value sign on sinx?
Why is there a absolute value sign on sinx? Does it have to do with the domain of cot x and sin x?
 
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It has to do with the domain of the logarithm. ##\ln \sin x ## isn't defined whenever ##\sin x \leq 0.##
 

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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