Question

can you please intergrate

Answer 1

\[\int\limits_{1}^{\infty}\theta x(1+x)^{-(1+\theta)}dx\]

Answer 2

will it be possible to use intergration by parts?

Answer 3

It substitution
let u = 1+x

Answer 4

thus x=u-1and dx=du

Answer 5

I mean this\[\int\limits_{2}^{\infty}\theta(u-1)u^{-1-\theta}\]

Answer 6

where did u get 2

Answer 7

since x is from 1 to infinity, u will be from x+1to infinity

Answer 8

b/c u=x+1

Answer 9

then will you do the rest?

Answer 10

i have \[-(\frac{ 2^{-\theta+1} }{ -\theta+1 }+\frac{ 2^{-\theta} }{ \theta })\]

Answer 11

I think the question is from 1 up to infinity, right?
Check once again

Answer 12

yes it is from 1 to infinity

Answer 13

the solution i typed bfore is already substituted with the limits of integration( infinity and 2)

Answer 14

yape correct

Answer 15

this\[\frac{ \theta }{ 2^{\theta}(1-\theta) }\]must be the answer

Answer 16

did u multiply with theta

Answer 17

I think you are incorrect while you integrate the integrand.

Answer 18

check this
\[\int\limits_{2}^{\infty}u ^{-\theta}-u ^{-(1+\theta)} du\]
\[(\frac{ u ^{-\theta+1} }{ -\theta+1 }+\frac{ u ^{-\theta} }{ \theta})\]

Answer 19

there it \[theta\] in front of the expression

Answer 20

it must be like this\[\int\limits_{2}^{\infty}({\theta}u^{-\theta}-{\theta}u^{-1-\theta})du\]

Answer 21

then i will have \[\theta(\frac{ 2^{-\theta+1} }{ -\theta+1}+\frac{ 2^{\theta} }{ theta})\]

Answer 22

now
\[\frac{ 2^{-\theta} (\theta+1)}{ -\theta+1 }\]

Answer 23

\[\frac{ \theta }{ 1-\theta }u^{1-\theta}-\frac{ \theta }{ -\theta }u^{-\theta}\]\[=\frac{ \theta }{ 1-\theta }u^{1-\theta}+u^{-\theta}\]\[=\frac{ \theta }{ 1-\theta }2^{1-\theta}+2^{-\theta}\]\[=(\frac{ 2{\theta} }{ 1-\theta }+1)\frac{ 1 }{ 2^{\theta} }\]\[=\frac{ \theta+1 }{ 2^{\theta} (1-\theta)}\]

Answer 24

yes i have it

hence from what we have

solve for theta
\[\frac{ \theta+1 }{ 2^{\theta} (1-\theta)}=x\]
i have this
\[\theta+1=2^{\theta}(1-\theta) x\]
\[\theta+1=2^{\theta}x-2^{\theta}x \theta \]
\[1=2^{\theta}x-2^{\theta}x \theta-\theta\]
then from here i don't know wat to do

Answer 25

@Zekarias