Question

integrate (lnx)(x^-(theta +1))dx from 1 to infinity

Answer 1

check your question

Answer 2

what u mean?

Answer 3

\[is it \int\limits x ^{-\theta+1}dx\]

Answer 4

\[\theta \int\limits_{1}^{\infty} (lnx)(x)\]

Answer 5

with x to the power of -(theta+1) dx

Answer 6

any ideas?

Answer 7

\[\int\limits_{1}^{\infty}\ln x \left( x \right)^{-\theta-1}dx\]
\[=[\ln x \frac{ x ^{-\theta} }{- \theta }] 1\to \infty -\int\limits_{1}^{\infty} \frac{ 1 }{ x }\frac{ x ^{-\theta} }{ -\theta }dx\]
\[=0+\frac{ 1 }{ \theta }\int\limits_{1}^{\infty}x ^{-\theta-1}dx=\frac{ 1 }{ \theta }\left[ \frac{ x ^{-\theta} }{-\theta } \right] 1 \to\infty \]
\[=\frac{ -1 }{ \theta ^{2} }\left( \frac{ 1 }{\infty }-\frac{ 1 }{1 } \right)=\frac{ -1 }{\theta ^{2} }\left( 0-1 \right)\]
\[=\frac{ 1 }{ \theta ^{2} }\]

Answer 8

what did u use to integrate it?

Answer 9

integrate by parts.
\[\ln 1=0\]

Answer 10

what are the parts?

Answer 11

thank you.

Answer 12

do u mind just writing out the parts u separated them into?

Answer 13

\[\ln x is first part ,and x ^{-\left( \theta+1 \right)} second part.\]

Answer 14

ok great thank you

Answer 15

\[\int\limits u v dx=u \int\limits v dx-\int\limits u \prime( \int\limits v dx )dx\]

Answer 16

awesome!

Answer 17

yw