MHB Definite Integration: $$\int^{\frac{\sqrt{5}+1}{2}}_{1}$$

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The integral $$\int^{\frac{\sqrt{5}+1}{2}}_{1}\frac{x^2+1}{x^4-x^2+1}\ln\left(x-\frac{1}{x}+1\right)dx$$ presents challenges in evaluation, prompting discussions on suitable methods. Participants suggest exploring substitution techniques and integration by parts, while others mention numerical approximation as a potential approach. Some have attempted partial fraction decomposition to simplify the integrand. The complexity of the logarithmic term adds to the difficulty, leading to considerations of series expansion. Overall, the thread emphasizes the need for innovative strategies to tackle this integral effectively.
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Evaluation of $$\int^{\frac{\sqrt{5}+1}{2}}_{1}\frac{x^2+1}{x^4-x^2+1}\ln\left(x-\frac{1}{x}+1\right)dx$$
 
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What methods are expected to use ? what have you tried so far ?
 

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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