Am i doing this question correctly.

[vkash]

Homework Statement

The integral of [x]+ln( (1+x)/(1-x) ) (lower limit -0.5 and upper limit is 0.5.)
here [x] show greater integer function means greatest integer less than equal to x. ex [2.3]=0.3,[1.7]=0.7,[-1.4]=0.6 etc.
//sorry that i don't know how to write it in better way

Homework Equations

millions so i can't write all of them here two of them is
integral of f(x) from a to b =integral of f(a+b-x) from a to b.
[1-x]=1-[x]
rest are in your mind.

The Attempt at a Solution

replace x by lower limit+ upper limit-x.
function will changed into [1-x]+ln( (1-x)/(1+x) )
replace [1-x] with 1-[x]. after little modification it will something like this I=integral with limits(1) -I
here I is the integral whose value i want to know.
finally the answer is 1/2 but this is wrong. correct answer as given in book is -1/2. My question is why my answer is wrong?
thanks for any kind of help.

vikash chandola
suport anna hazare(only for indians)

 

[ArcanaNoir]

You have the greatest integer function wrong. example: [2.3]=2
As for the rest of the problem, I'm not sure how to do it.

 

[SammyS]

$\displaystyle \int_{-0.5}^{0.5}\lfloor x \rfloor dx$ should be a piece of cake.

For -1 ≤ x < 0 , $\displaystyle \lfloor x \rfloor = -1$.

For 0 ≤ x < 1 , $\displaystyle \lfloor x \rfloor = 0$.​

For $\displaystyle \ln\left(\frac{1+x}{1-x}\right)\,,$ use the rule for the log of a quotient. Then break-up the integral accordingly.

Do you know the anti-derivative of ln(x) ?

 

[vkash]

You have the greatest integer function wrong. example: [2.3]=2
As for the rest of the problem, I'm not sure how to do it.

Oh man. what a foolish work i did. I understand greater integer function as fraction part function. I also applly wrong property that's why i got wrong answer.After thanks you tel me my mistake.Hey sammy what are you writing. this is seeming lengthy method. see how i solve it. thing i do wrong is understand greatest integer function as fraction part function. So the property [-x]=1-[x] is wrong. Correct one is [-x]=-1-[x]. now i will get correct answer.

anti derivative of ln(x) is xln(x)-x+C

 

[SammyS]

The problem has [x]. Why are you working with [-x] ?

 

[vkash]

The problem has [x]. Why are you working with [-x] ?

oh one more sorry. Actually the place where i write [1-x] that is [-x] =-1-[x].
I do this mistake however my trick is correct. thanks for showing me my mistake. If you still say how yo write [-x] then something is wrong. either i am doing one more mistake or you do something wrong.