Find-1-dx-x-2-x-1-2-x-x-1-x-1-Where-denote-the-integer-part-

Question Number 140827 by mathsuji last updated on 13/May/21

F i n d

1 . \int \frac{d x}{{[x]}^{2}}, x ⩾ 1

2 . \int \frac{{[x]}^{λ}}{x^{λ + 1}}, x ⩾ 1

W h e r e [*] d e n o t e t h e i n t e g e r p a r t

Answered by mathmax by abdo last updated on 13/May/21

if youmean \int_{1}^{\infty} \frac{dx}{{[x]}^{2}} let I = \int_{1}^{\infty} \frac{dx}{{[x]}^{2}} \Rightarrow

I = \sum_{n = 1}^{\infty} \int_{n}^{n + 1} \frac{dx}{n^{2}} = \sum_{n = 1}^{\infty} \frac{1}{n^{2}} = \frac{π^{2}}{6}

if I = \int_{1}^{x} \frac{dt}{{[t]}^{2}} \Rightarrow I = \sum_{n = 1}^{[x] - 1} \int_{n}^{n + 1} \frac{dt}{n^{2}} = \sum_{n = 1}^{[x] - 1 w} \frac{1}{n^{2}}

Commented by mathsuji last updated on 16/May/21

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