Improper integrals For x+1/sqrt(x^2+2x)
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Improper integrals For x+1/sqrt(x^2+2x)

[From: ] [author: ] [Date: 12-11-01] [Hit: ]
com/r… http://esv.scripturetext.com/r… http://bible.cc/1_john/1-5.......
The integral from 0 to 1 of x+1/sqrt(x^2+2x). I'm looking at this and it seems like a u substitution would work but I can't seem to find one that works. I tried u = x^2 but things started getting ugly. Is u substitution viable or should I try another method? Anyone got any ideas?

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Let u = x^2 + 2x. Then du = (2x + 2) dx = 2*(x + 1) dx. The integral then becomes

integral((1/2)*u^(-1/2) du) = (1/2)*u^(1/2) / (1/2) + C = (x^2 + 2x)^(1/2) + C.

Now evaluate from 0 to 1 to get (1 + 2)^(1/2) - 0 = 3^(1/2) = sqrt(3).

I assumed that the original expression was (x + 1)/sqrt(x^2 + 2x) and not

x + (1 / sqrt(x^2 + 2x)). If you meant the latter then the solution is

(1/2) + 2*arcsinh(1/sqrt(2)), but since I think that you meant the former

I won't bother showing the steps unless you request them.

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http://www.pravoslavie.ru/engl… http://www.youtube.com/watch?v… http://esv.scripturetext.com/r… http://esv.scripturetext.com/r… http://bible.cc/1_john/1-5.htm

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