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% !!! IMAGES START HERE !!!

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$\displaystyle \int \frac{dx}{x^2 \sqrt{4- x^2}} $%
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{\newpage\clearpage
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$\displaystyle \int \frac{3x^2+x-2}{x^2(x-2)} dx $%
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{\newpage\clearpage
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$\displaystyle
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{\newpage\clearpage
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$x = 2 sen(u) \Rightarrow dx = 2 cos(u) du \ ,
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{\newpage\clearpage
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$\displaystyle \int \frac{dx}{x^2 \sqrt{4- x^2}} = 
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{\newpage\clearpage
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$\Longrightarrow \displaystyle \int \frac{dx}{x^2 \sqrt{4- x^2}} = 
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{\newpage\clearpage
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$\displaystyle cotg(u) = \frac{\cos(u)}{sen(u)} =
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{\newpage\clearpage
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$\displaystyle \int \frac{dx}{x^2 \sqrt{4- x^2}} = 
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{\newpage\clearpage
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$\displaystyle \frac{3x^2 + x -2}{x^2(x-2)} = \frac{A}{x} +
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{\newpage\clearpage
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$\displaystyle 3x^2 + x -2 = A(x^2 - 2x) + B(x-2) + Cx^2$%
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{\newpage\clearpage
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$\displaystyle 3x^2 + x -2 = (A + C)x^2 + (B-2A)x - 2B$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline333}%
$\displaystyle \left\{ \begin{array}{ccc}
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{\newpage\clearpage
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$\displaystyle B = 1 \Rightarrow 2A = B-1 = 0 \Rightarrow A = 0 $%
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{\newpage\clearpage
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$\displaystyle C = 3 - A = 3 - 0 = 3$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline339}%
$\displaystyle \frac{3x^2 + x -2}{x^2(x-2)} = \frac{1}{x^2} +
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline341}%
$\displaystyle \int \frac{3x^2+x-2}{x^2(x-2)} dx  =
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3 \ln |x-2| - \frac{1}{x} + C \ , \  C \in {\mathbb{R}}$%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline345}%
$\displaystyle \int_1^{\ell} 3 x^{-3/2} dx = 3 \left[
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline347}%
$\displaystyle \int_1^{\infty} \frac{3}{x \sqrt{x}} dx =
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{\newpage\clearpage
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$y=x^2$%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline353}%
$\displaystyle x^2 = 4x-x^2 \Leftrightarrow 2x^2 - 4x = 0
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline355}%
$\displaystyle \Rightarrow
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{\newpage\clearpage
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$\displaystyle y_1 = x_1^2 = 0 \ , \ y_2 = x_2^2 = 4$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline359}%
$\displaystyle A =  \int_{0}^{2} \left[
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline361}%
$\displaystyle A =  \int_0^2 ( 4x-2x^2 ) dx =
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2 (2)^2 - \frac{2 (2)^3}{3} - 0 + 0 = $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline363}%
$\displaystyle A = 8 - \frac{2(8)}{3} = \left( 1 - \frac{2}{3} 
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{\newpage\clearpage
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$\displaystyle g(x) = x^2-1$%
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{\newpage\clearpage
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$f(x) = 2 \sqrt{5x+1}$%
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{\newpage\clearpage
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$x$%
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{\newpage\clearpage
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$y$%
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{\newpage\clearpage
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$f$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline375}%
$5x + 1 = 0 \Rightarrow x = -1/5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline377}%
$g$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline379}%
$x^2 = 1 , x > 0 \Rightarrow x=1$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline381}%
$\displaystyle A = \int_{-1/5}^1 2\sqrt{5x+1}dx +
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
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$ y = x^2 -1 \Rightarrow x = \sqrt{y+1} = x_g$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline385}%
$\displaystyle
y = 2\sqrt{5x+1} \Rightarrow (y/2)^2 = 5x + 1 \Rightarrow
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\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline387}%
$\displaystyle A = \int_0^8 (x_g - x_f) dy=
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\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline389}%
$\displaystyle F(x) = \int_{1}^x f(t) dt \ , \ 1 \leq x \leq 4$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline391}%
$y = f(x)$%
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{\newpage\clearpage
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$A_1$%
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{\newpage\clearpage
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$A_2$%
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{\newpage\clearpage
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$A_3$%
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{\newpage\clearpage
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$F(1), F(2), F(3), F(4)$%
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{\newpage\clearpage
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$F$%
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{\newpage\clearpage
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$\displaystyle F(1) = \int_{1}^{1} f(t)dt = 0$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline407}%
$\displaystyle F(2) = F_1 - 4 = -4$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline409}%
$\displaystyle F(3) = F_2 + 2 = -2$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline411}%
$\displaystyle F(4) = F(3) -1 = -3$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline413}%
$\displaystyle F'(x) = \frac{d}{dx} \left[ \int_{1}^x f(t) dt \right] =
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{\newpage\clearpage
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$F'(x)= f(x) < 0 $%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline417}%
$(1,2) \bigcup (3,4)
\Longrightarrow F(x) $%
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{\newpage\clearpage
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$(1,2) \bigcup (3,4)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline421}%
$F'(x) = f(x) > 0 $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline423}%
$(2,3) \Longrightarrow F(x) $%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline425}%
$(2,3)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline427}%
$(-10,0)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline429}%
$(0,0)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline431}%
$(-2,2)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline433}%
$\displaystyle V_1(-10,0) \ , \ V_2(0,0) \Longrightarrow 2a = 10 
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\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline435}%
$X$%
\lthtmlinlinemathZ
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{\newpage\clearpage
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$C(-5,0)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline439}%
$\displaystyle \frac{(x-(-5))^2}{a^2} + \frac{(y-0)^2}{b^2} = 1
\displaystyle \frac{(x+5)^2}{5^2} + \frac{y^2}{b^2} = 1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline441}%
$(x,y) = (-2,2)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline443}%
$\displaystyle \frac{(-2+5)^2}{5^2} + \frac{2^2}{b^2} = 1 \Rightarrow 
\frac{9}{25} + \frac{4}{b^2} = 1$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline445}%
$\displaystyle
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\Rightarrow \frac{1}{b^2} = \frac{4}{25}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline447}%
$b = 5/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline449}%
$\displaystyle  \frac{(x+5)^2}{25} + \frac{4y^2}{25} = 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline451}%
$a^2 = b^2 + c^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline453}%
$\displaystyle  5^2 = \left(\frac{5}{2} \right)^2 + c^2 \Rightarrow 
c^2 = \frac{25(3)}{2} \Rightarrow c = \frac{5 \sqrt{3}}{2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline455}%
$\displaystyle F_1 = \left( -5 - \frac{5 \sqrt{3}}{2},0 \right) \ , \ 
F_2 = \left( -5 + \frac{5 \sqrt{3}}{2},0 \right)  $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline457}%
$a,b,c$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}


\end{document}
