1. ## improper integral

hey everyone, got 2 questions.

the first one asks for determining all values of p for which dx/(x-p) is improper evaluated from 1 to 2.

another question asks about finding an equation for the integral curve that passes through the point (2,1) for the differential equation y'=y/2x and how do you go about graphing y'=y/2x?

cheers!

2. a) $\displaystyle \int_1^2 {\frac{1} {{x - p}}dx} = \ln \left( {2 - p} \right) - \ln \left( {1 - p} \right)$

we know that the domain the logarythm function is to: $\displaystyle x>0$ then to this integral con be improper: $\displaystyle 2 - p \leqslant 0 \vee 1 - p \leqslant 0 \Rightarrow \therefore p \geqslant 1$

b) Maybe could be a better idea, find Y

$\displaystyle y'(x) = \frac{{y(x)}} {{2x}} \Leftrightarrow \frac{{y'(x)}} {{y(x)}} = \frac{1} {{2x}}$

Integrating whit respecte X, we have: $\displaystyle \int {\frac{{y'(x)}} {{y(x)}}dx} = \frac{1} {2}\int {\frac{1} {x}dx} \Leftrightarrow \ln \left( {y(x)} \right) = \frac{1} {2}\ln x + C$

Hence: $\displaystyle y(x)=Ae^{\sqrt{x}}$

But curve passes through the point (2,1) then $\displaystyle y(2) = 1 \Leftrightarrow 1 = Ae^{\sqrt 2 }$ $\displaystyle \therefore y(x) = \exp \left( {x - \sqrt 2 } \right)$