Does anyone know how to do these? Have a whole sheet of practice

>>8449968
Holy shit is it possible to be this dumb?

Sage, belongs is sqt

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>>8449968
Stupid questions thread?
How was this integrated?

>>8450031
The integral of 1/x is lnx

>>8449968
If you take the series solution of
[eqn] \sqrt{x^2 + \epsilon} = \mathrm{e}^x - 1[/eqn]
then you get
[eqn]x = \epsilon^{\frac{1}{3}} - \frac{7}{36} \epsilon^{\frac{2}{3}} + \frac{13}{432} \epsilon - \frac{733}{699840} \epsilon^{\frac{4}{3}} - \frac{14567}{12597120} \epsilon^{\frac{5}{3}} + \frac{3139}{8709120} \epsilon^{2} - \frac{2025773}{171421608960} \epsilon^{\frac{7}{3}} - \frac{3386640461}{123423558451200} \epsilon^{\frac{8}{3}}+ O( \epsilon^{3} ) [/eqn]

Just cut off everything but the first term.

>>8449968
To solve as a linear approximation set [math]x = A + B \epsilon[/math], expand as series and solve for A and B.

Square both sides to make life easier, you'll have to filter out some of the solutions later.
[math]\displaystyle x^2+\epsilon = e^{2 x} - 2 e^x + 1[/math]
[math]\displaystyle A^2 + (1 + 2 A B) \epsilon + \cdots = (e^A-1)^2 + 2 B e^A (e^A-1) \epsilon + \cdots[/math]

Don't care enough to solve the problem fully but the solution will be something like
[math]\displaystyle x = A + \frac{1}{2 A} \frac{1}{e^A-1} \epsilon[/math]
where
[math]\displaystyle A = -1-\mathrm{W}(-\frac{1}{e})[/math]
W(x) is the Lambert Omega function, which is multi-valued so there are many solutions.

>>8450086
How did you get this?

>>8451138
Very carefully

>>8449968
What the heck is a "leading-order approximation" ?

>>8450086
If the problem were
[math]\sqrt{x^2 + t^3}[/math] you'd have an x an analytic function of t ? Interesting.