∫ 0 1 a k x k d x =

\int_{0}^{1} a^{k} x^{k} d x =

$\lim_{n \to \infty} \frac{a^{k} (1 + 2^{k} + 3^{k} \dots + n^{k})}{n^{k + 1}}$
$\lim_{n \to \infty} \frac{a^{k} + a^{k} + \dots + a^{k}}{n^{k + 1}}$
$\lim_{n \to \infty} \frac{1}{n} \sum {(\frac{r}{n})}^{k}$
$\lim_{n \to \infty} \frac{1}{n} \sum {(\frac{2 r}{n})}^{k}$

Solution

$\int_{0}^{1} a^{k} x^{k} d x = a^{k} {[\frac{x^{k + 1}}{k + 1}]}_{0}^{1}$
$= \frac{a^{k}}{k + 1}$
Now, $\lim_{n \to \infty} \frac{a^{k} (1 + 2^{k} + 3^{k} \dots + n^{k})}{n^{k + 1}}$ $= a^{k} \lim_{n \to \infty} \sum_{r = 1}^{n} \frac{r^{k}}{n \cdot n^{k}} = a^{k} \int_{0}^{1} x^{k} d x = \frac{a^{k}}{k + 1}$

Asked in: AP EAMCET 2022 (04 Jul Shift 1)

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