Definite Integral: Limit of a Summation

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Homework Statement



Hi guys, i have a exercise of the limit of a summation that is the formal definition of definite integral and i need resolve and explain, but i can't resolve for the rational exponent, for this, need help, thanks in advance.

Homework Equations



\lim_{n \rightarrow \infty} \sum_{i=1}^{n} {(1+\frac{2}{n}(i-0.3))^{\frac{7}{5}}\frac{2}{n}

The Attempt at a Solution



I can solve this expretion but with a integer exponent, not with a rational exponent.
 
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Help me, please.
 
That really doesn't look like a Riemann sum to me. Were you given that sum?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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