Question

can someone help me handle this integral

Answer 1

For which p converges\[\int\limits_{1}^{∞}\frac{ \ln(x) }{ x^p }\]

Answer 2

x>1

Answer 3

x \(\ge\) 1?

Answer 4

use integration by parts

Answer 5

You mean p?

Answer 6

i think ln(1) is also legit right?

Answer 7

yes yes sorry ... p=1, for some x, log(x) > 1 for x>e. should diverge for p=1. it converges for p>1

Answer 8

\[\int\limits\limits_{}^{}\frac{ \ln(x) }{ x^p }=\left[ \frac{ x^{p-1}-x^{p-1}\ln(x)(p+1) }{ (p-1)^2 } \right]\]

Answer 9

\[\lim_{b \rightarrow ∞}\left[ \frac{ x^{p-1}-x^{p-1}\ln(x)(p+1) }{ (p-1)^2 } \right]^b_1\]

Answer 10

\[\lim_{b \rightarrow ∞} \frac{ b^{p-1}-b^{p-1}\ln(b)(p+1) }{ (p-1)^2 } - \frac{ 1^{p-1}-1^{p-1}\ln(1)(p+1) }{ (p-1)^2 }\]

Answer 11

\[ \int_1^\infty \frac{\log x }{x^p}dx = \left[\frac{\log(x) x^{1- p}}{1-p}\right]_1^\infty - \frac{1}{1-p}\int_1^\infty \frac 1 x \cdot x^{1-p}dx = 0 - ... \]

Answer 12

thank you

Answer 13

yw