Question

integration by substitution its really tough try once..!

Answer 1

\[\int\limits_{0}^{1}\frac{ \log_{e}(x+1) }{ x } dx \]

Answer 2

i promise a medal and would be really a fan of the person who solves it......!

Answer 3

try
\[u=\ln(x+1)\]

Answer 4

that's no good

Answer 5

ooh i see ....

Answer 6

there must be some gimmick with the limits of integration then
you are not going to get a nice closed form for this integral

Answer 7

Here's an excellent article on the dilogarithm function.

http://maths.dur.ac.uk/~dma0hg/dilog.pdf

Enjoy your Summer reading!

Answer 8

i tried all possible substitutions but it was just going round and round

Answer 9

@tkhunny its easy to do by expanding the function i am much concerned with a solution by substitution ....

Answer 10

the answer i got from the expansion of function is\[\pi ^{2}/12\]

Answer 11

You may be concerned as you wish. It can't be done. Why do you think it can? Did you read the article, already?

Answer 12

i think it could be done because i believe the only thing impossible in the world is the possibility of impossibility @tkhunny

Answer 13

That's simply a very incorrect understanding of mathematical proof. A mathematical proof is not a mere suggestion that it MAY be done, if only we are sufficiently clever. A mathematical proof has the power to say it CANNOT be done. It has nothing at all to do with belief or philosophy.

Techniques of integration introduced in standard textbooks of of very limited use out in the real world. They are great for a couple dozen practice problems, but only by luck will you find such a technique helpful with a real world problem.

Answer 14

@harsh314, By expanding, are you referring to a power series?

Answer 15

Well, a power series isn't going to give you \(\pi^{2}/12\).

Answer 16

yes that's exactly what i referred @SithsAndGiggles..

Answer 17

@tkhunny first of all its not a philosophy. second thing is that mathematical proofs never did scale my imagination. Third thing is i am posting the solution by expansion. The fourth thing is u are not lucky enough to have a chance to apply integration to real life. The last thing is the world is much bigger than as much you see.

Answer 18

Philosophy: Yes it is. You're in denial.

Scale: It's not mathematic's fault. It's your mind.

Posting: Feel free.

Application: Integration has many applications, just not much for the basic techniques in the book.

World: No, I pretty much have a handle on the scope. I've been around with my eyes open long enough.

Good try, though.

Answer 19

philosophy it is not and you are adamant ....................... mathematics can't measure abstract nouns................. i am posting not to satisfy you but shatter some fancies ..... some people are blind even with open eyes.........

Answer 20

I'm perfectly fine with your pursuits. This is how we get crackpots in our society. They have a wonderful dream that flies in the face of truth and knowledge. If you choose to be that way, that is your choice. I made a small effort to talk you out of this destructive course, but the choice is ultimately yours.

Have a great day.

Answer 21

\[\int\limits_{0}^{1}\frac{ \log_{e}(1+x) }{ x }dx = \int\limits_{0}^{1}\frac{ x-\frac{ x ^{2} }{ 2 }+\frac{ x ^{3} }{ 3 }..... }{ x}\]\[=\int\limits_{0}^{1}1-\frac{ x }{ 2 }+\frac{ x ^{2}}{ 3 }.............\] integrating it \[1-\frac{ 1 }{ 4}+\frac{ 1 }{ 9 }................ \infty\]

Answer 22

remember one thing only dreams change the reality and sometimes correct the commonly accepted truths and the so called knowledge.......! have good week month year and centuries

Answer 23

Right. A power series and someone else's work that told you what the sum was. Like I said, a power series won't do it. It takes a power series and a little extra push.

Notice how I carefully avoided the adjectives you inserted. I spoke of truth and knowledge, only. I absolutely agree that "commonly accepted truths" and "so-called knowledge" should be questioned. Truth and knowledge, though, should be heeded and not abandoned.

Have a great eon, I guess.

Answer 24

@tkhunny friend i would keep it simple this time if u don't know how to solve it by substitution then try .... if you don't want to do it .... then plz don't try to conceal some inefficiency by generalising it as impossibility...... the second thing is what you are saying someone else's work i guess i am that someone................. the adjectives you are talking about is with reference to your comments stating the question can't be solved ..... that was the commonly accepted the truth ...... what i shatter.....the last thing i would not live as longer as an eon......... hope my name does .....

Answer 25

Good luck with that.

Answer 26

thanks....!!!!! and sorry if i did abuse you anytime in the discourse...

Answer 27

\[1-\frac{ 1 }{ 4}+\frac{ 1 }{ 9 }-\cdots=\sum_{k=1}^\infty\frac{(-1)^{n+1}}{n^2}~~\text{is a convergent sum, not divergent.}\]

Answer 28

No, no. Quite a lively debate. When you have the solution to the general quintic equation, you let me know. I'd be delighted to review it.