MHB Can Integration by Parts Solve This Tricky Question?

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The discussion centers on solving a challenging integration problem using integration by parts. The user proposes letting u = x^3 and dv = x^2√(x^3 + 1) for the integration by parts approach. An alternative method suggested involves using substitution with t = x^3 + 1. Participants are encouraged to provide assistance and insights on the effectiveness of these methods. The conversation highlights the complexity of the integration question and the various strategies that can be employed.
jaychay
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Can you help me with this question ?
I am really struck with this question.
Thank you in advance.
 
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Using integration by parts, I would let

$u = x^3$ and $dv=x^2\sqrt{x^3+1}$Alternatively, one could use the same setup for the method of substitution letting $t= x^3+1$
 

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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