# LaTeX2HTML 2002-2-1 (1.70)
# Associate images original text with physical files.


$key = q/Cin{{mathbb{R}{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.gif"
 ALT="$C \in {\mathbb{R}}$">|; 

$key = q/a^2=b^2+c^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.gif"
 ALT="$a^2 = b^2 + c^2$">|; 

$key = q/displaystyletan(u)=xslash3Rightarrowsec(u)=sqrt{1+(xslash3)^2}=frac{sqrt{9+x^2}}{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="385" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.gif"
 ALT="$\displaystyle \tan(u) = x/3 \Rightarrow \sec(u) = \sqrt{1 + (x/3)^2} =
\frac{\sqrt{9 + x^2}}{3}$">|; 

$key = q/displaystyley=x+2Rightarrowx=y-2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.gif"
 ALT="$\displaystyle y = x + 2 \Rightarrow x = y-2 $">|; 

$key = q/displaystyleF'(x)=frac{d}{dx}left[int_{-1}^xf(t)dtright]=f(x),xin(-1,3);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="350" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.gif"
 ALT="$\displaystyle F'(x) = \frac{d}{dx} \left[ \int_{-1}^x f(t) dt \right] =
f(x)  ,  x \in (-1,3)$">|; 

$key = q/a,b,c;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.gif"
 ALT="$a,b,c$">|; 

$key = q/displaystyleC(4,0),F(1,0)Longrightarrowc=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.gif"
 ALT="$\displaystyle C(4,0)  ,  F(1,0) \Longrightarrow c = 3 $">|; 

$key = q/X;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img58.gif"
 ALT="$X$">|; 

$key = q/displaystyle4x=5-x^2Rightarrowx^2+4x-5=(x+5)(x+1)=0Rightarrowx=1,y=4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="488" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.gif"
 ALT="$\displaystyle 4x = 5 - x^2 \Rightarrow x^2 + 4x - 5 = (x+5)(x+1) = 0
\Rightarrow x=1, y=4$">|; 

$key = q/x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img28.gif"
 ALT="$x$">|; 

$key = q/displaystyle2x^2-x+1=A(x^2-2x+1)+B(x^2-1)+C(x+1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.gif"
 ALT="$\displaystyle 2x^2 - x + 1 = A(x^2 - 2x + 1) + B(x^2-1) + C(x+1)$">|; 

$key = q/displaystyle4^2=b^2+3^2Longrightarrowb^2=7;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.gif"
 ALT="$\displaystyle 4^2 = b^2 + 3^2 \Longrightarrow b^2 = 7$">|; 

$key = q/F'(x)=f(x)>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.gif"
 ALT="$F'(x) = f(x) &gt; 0 $">|; 

$key = q/displaystyleF(3)=F(2)+1+frac{1}{2}=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.gif"
 ALT="$\displaystyle F(3) = F(2) + 1 + \frac{1}{2} = 1$">|; 

$key = q/displaystylefrac{(x-4)^2}{16}+frac{y^2}{7}=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.gif"
 ALT="$\displaystyle \frac{(x-4)^2}{16} + \frac{y^2}{7} = 1$">|; 

$key = q/(1,3);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.gif"
 ALT="$(1,3)$">|; 

$key = q/(1,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.gif"
 ALT="$(1,0)$">|; 

$key = q/F;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img43.gif"
 ALT="$F$">|; 

$key = q/displaystyleint_5^{infty}(3x+1)^{-3slash2}dx=lim_{ellrightarrowinfty}frac{2}{3}lac{1}{sqrt{3ell+1}}right)=frac{2}{3}cdotfrac{1}{4}=frac{1}{6};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="444" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.gif"
 ALT="$\displaystyle \int_5^{\infty} (3x+1)^{-3/2} dx =
\lim_{\ell \rightarrow \infty}...
...frac{1}{\sqrt{3\ell + 1}} \right) = \frac{2}{3} \cdot \frac{1}{4} = \frac{1}{6}$">|; 

$key = q/displaystyleF(-1)=int_{-1}^{-1}f(t)dt=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.gif"
 ALT="$\displaystyle F(-1) = \int_{-1}^{-1} f(t)dt = 0$">|; 

$key = q/displaystyleintfrac{2x^2-x+1}{(x+1)(x-1)^2}dx=intleft(frac{1}{x+1}+frac{1}{x-1}+(x-1)^{-2}right)dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="455" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.gif"
 ALT="$\displaystyle \int \frac{2x^2 - x + 1}{(x+1)(x-1)^2} dx =
\int \left( \frac{1}{x+1} + \frac{1}{x-1} + (x-1)^{-2} \right) dx $">|; 

$key = q/y=x+2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.gif"
 ALT="$y = x+2$">|; 

$key = q/displaystyleint_5^{ell}(3x+1)^{-3slash2}dx=int_{16}^{3ell+1}u^{-3slash2}frac{du}c{1}{3}left[frac{u^{-1slash2}}{-1slash2}right]_{16}^{3ell+1}=;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.gif"
 ALT="$\displaystyle \int_5^{\ell} (3x+1)^{-3/2} dx = \int_{16}^{3\ell + 1}
u^{-3/2} \frac{du}{3} = \frac{1}{3} \left[ \frac{u^{-1/2}}{-1/2}
\right]_{16}^{3 \ell +1}=$">|; 

$key = q/displaystylefrac{16}{a^2}+frac{0^2}{b^2}=1Longrightarrowa=4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.gif"
 ALT="$\displaystyle \frac{16}{a^2} + \frac{0^2}{b^2} = 1 \Longrightarrow a = 4$">|; 

$key = q/displaystyleF(2)=F(1)+frac{1}{2}=frac{-1}{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.gif"
 ALT="$\displaystyle F(2) = F(1) + \frac{1}{2} = \frac{-1}{2} $">|; 

$key = q/displaystyle2x^2-x+1=(A+B)x^2+(C-2A)x+A-B+C;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="401" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.gif"
 ALT="$\displaystyle 2x^2 - x + 1 = (A + B)x^2 + (C-2A)x + A - B + C$">|; 

$key = q/displaystylefrac{2x^2-x+1}{(x+1)(x-1)^2}=frac{1}{x+1}+frac{1}{x-1}+frac{1}{(x-1)^2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.gif"
 ALT="$\displaystyle \frac{2x^2 - x + 1}{(x+1)(x-1)^2} = \frac{1}{x+1} +
\frac{1}{x-1} + \frac{1}{(x-1)^2} $">|; 

$key = q/displaystyleA=frac{4}{3}left(1^{3slash2}-0^{3slash2}right)+left(frac{2}{3}cdot4^lash2}-0-0-frac{2}{3}cdot1^{3slash2}+frac{3^2}{2}+2(-3)right);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="503" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.gif"
 ALT="$\displaystyle A = \frac{4}{3} \left( 1^{3/2} - 0^{3/2} \right) +
\left( \frac{2...
...ot 4^{3/2} - 0 - 0 - \frac{2}{3} \cdot 1^{3/2} +
\frac{3^2}{2} + 2(-3) \right) $">|; 

$key = q/displaystyley_1=x_1+2=2,y_2=x_2+2=-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.gif"
 ALT="$\displaystyle y_1 = x_1 + 2 = 2  ,  y_2 = x_2 + 2 = -1$">|; 

$key = q/(1,3)LongrightarrowF(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.gif"
 ALT="$(1,3) \Longrightarrow F(x) $">|; 

$key = q/displaystyleRightarrowx^2+3x=x(x+3)=0Rightarrowx_1=0,x_2=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.gif"
 ALT="$\displaystyle \Rightarrow
x^2 + 3x = x(x+3) = 0 \Rightarrow x_1 = 0  ,  x_2 = 3$">|; 

$key = q/displaystylefrac{2x^2-x+1}{(x+1)(x-1)^2}=frac{A}{x+1}+frac{B}{x-1}+frac{C}{(x-1)^2}=frac{A(x-1)^2+B(x-1)(x+1)+C(x+1)}{(x+1)(x-1)^2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="669" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.gif"
 ALT="$\displaystyle \frac{2x^2 - x + 1}{(x+1)(x-1)^2} = \frac{A}{x+1} +
\frac{B}{x-1} + \frac{C}{(x-1)^2} = \frac{A(x-1)^2 + B(x-1)(x+1) + C(x+1)}
{(x+1)(x-1)^2}$">|; 

$key = q/displaystyle2A=C+1=2RightarrowA=1,RightarrowB=2-A=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.gif"
 ALT="$\displaystyle 2A = C + 1 = 2 \Rightarrow A = 1  , \
\Rightarrow B = 2 - A = 1$">|; 

$key = q/y^2=x+4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.gif"
 ALT="$y^2 = x+4$">|; 

$key = q/x=3tan(u)Rightarrowdx=3sec^2(u)du,-pislash2<u<pislash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.gif"
 ALT="$x = 3 \tan(u) \Rightarrow dx = 3 \sec^2(u) du  ,
 -\pi/2 &lt; u &lt; \pi/2$">|; 

$key = q/displaystylex+4=(x+2)^2Rightarrowx^2+4x+4=x+4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.gif"
 ALT="$\displaystyle x+4 = (x+2)^2 \Rightarrow x^2 + 4x + 4 = x+ 4 $">|; 

$key = q/(4,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.gif"
 ALT="$(4,0)$">|; 

$key = q/F(-1),F(1),F(2),F(3);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.gif"
 ALT="$F(-1), F(1), F(2), F(3)$">|; 

$key = q/displaystyleA=frac{4}{3}+frac{16}{3}-frac{2}{3}+frac{9}{2}-6=frac{9}{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.gif"
 ALT="$\displaystyle A = \frac{4}{3} + \frac{16}{3} - \frac{2}{3}
+ \frac{9}{2} - 6 =\frac{9}{2}$">|; 

$key = q/displaystyleV=piint_0^1left((5-x^2)^2-(4x)^2right)dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.gif"
 ALT="$\displaystyle V = \pi \int_0^1 \left( (5-x^2)^2 - (4x)^2 \right) dx $">|; 

$key = q/displaystyleintfrac{2x^2-x+1}{(x+1)(x-1)^2}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.gif"
 ALT="$\displaystyle \int \frac{2x^2-x+1}{(x+1)(x-1)^2} dx $">|; 

$key = q/displaystyle2C=2RightarrowC=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img13.gif"
 ALT="$\displaystyle 2C = 2 \Rightarrow C = 1$">|; 

$key = q/displaystyleA=int_{-4}^{-3}2sqrt{x+4}dx+int_{-3}^0left[sqrt{x+4}-(x+2)right]dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.gif"
 ALT="$\displaystyle A = \int_{-4}^{-3} 2 \sqrt{x+4} dx + \int_{-3}^0
\left[ \sqrt{x+4} - (x+2) \right] dx $">|; 

$key = q/(-1,1)LongrightarrowF(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.gif"
 ALT="$(-1,1) \Longrightarrow F(x) $">|; 

$key = q/displaystyleint_5^{+infty}(3x+1)^{-3slash2}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.gif"
 ALT="$\displaystyle
\int_5^{+\infty} (3x+1)^{-3/2} dx$">|; 

$key = q/C(4,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.gif"
 ALT="$C(4,0)$">|; 

$key = q/displaystyley^2=x+4Rightarrowx=y^2-4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="191" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.gif"
 ALT="$\displaystyle y^2 = x+4 \Rightarrow x = y^2 - 4$">|; 

$key = q/(-1,1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.gif"
 ALT="$(-1,1)$">|; 

$key = q/displaystylefrac{(x-4)^2}{a^2}+frac{(y-0)^2}{b^2}=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.gif"
 ALT="$\displaystyle \frac{(x-4)^2}{a^2} + \frac{(y-0)^2}{b^2} = 1$">|; 

$key = q/displaystyleA=int_{-1}^2left(y-2-(y^2-4)right)dy=int_{-1}^2(y+2-y^2)dy=left[frac=2+3-frac{8}{3}-left(frac{1}{2}-2+frac{1}{3}right)=frac{9}{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="802" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.gif"
 ALT="$\displaystyle A = \int_{-1}^2 \left( y-2 - (y^2 - 4)\right) dy =
\int_{-1}^2 (y...
...2 + 3 -\frac{8}{3} - \left( \frac{1}{2} - 2 +
\frac{1}{3} \right) = \frac{9}{2}$">|; 

$key = q/y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.gif"
 ALT="$y$">|; 

$key = q/displaystyleint_5^{infty}(3x+1)^{-3slash2}dx=lim_{ellrightarrowinfty}int_5^{ell}(3x+1)^{-3slash2}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="344" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.gif"
 ALT="$\displaystyle \int_5^{\infty} (3x+1)^{-3/2}dx = \lim_{\ell \rightarrow
\infty} \int_5^{\ell} (3x+1)^{-3/2} dx$">|; 

$key = q/displaystyleintfrac{dx}{sqrt{9+x^2}}=lnleft|frac{sqrt{9+x^2}}{3}+frac{x}{3}right|+C,Cin{{mathbb{R}{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.gif"
 ALT="$\displaystyle \int \frac{dx}{\sqrt{9 + x^2}} = \ln \left\vert
\frac{\sqrt{9 + x^2}}{3} + \frac{x}{3} \right\vert + C  ,  C \in {\mathbb{R}}$">|; 

$key = q/y=f(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.gif"
 ALT="$y = f(x)$">|; 

$key = q/displaystyleA=2left[frac{(x+4)^{3slash2}}{3slash2}right]_{-4}^{-3}+left[frac{(x+4)^{3slash2}}{3slash2}-frac{x^2}{2}-2xright]_{-3}^0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.gif"
 ALT="$\displaystyle A = 2 \left[ \frac{(x+4)^{3/2}}{3/2} \right]_{-4}^{-3} +
\left[ \frac{(x+4)^{3/2}}{3/2} - \frac{x^2}{2} - 2x \right]_{-3}^0 $">|; 

$key = q/F'(x)=f(x)<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.gif"
 ALT="$F'(x)= f(x) &lt; 0 $">|; 

$key = q/displaystylefrac{-2}{3}left[frac{1}{sqrt{3ell+1}}-frac{1}{sqrt{16}}right]=frac{2}{3}left(frac{1}{4}-frac{1}{sqrt{3ell+1}}right);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.gif"
 ALT="$\displaystyle \frac{-2}{3} \left[ \frac{1}{\sqrt{3\ell + 1}} -
\frac{1}{\sqrt{16}} \right] = \frac{2}{3} \left( \frac{1}{4} - \frac{1}
{\sqrt{3\ell + 1}} \right)$">|; 

$key = q/displaystyleV=piint_0^4left(frac{y}{4}right)^2dy+piint_4^5left(sqrt{5-y}right)^2dy=frac{pi}{16}int_0^4y^2dy+piint_4^5(5-y)dy;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="565" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.gif"
 ALT="$\displaystyle V = \pi \int_0^4 \left( \frac{y}{4} \right)^2 dy +
\pi \int_4^5 \...
...t{5 - y}  \right)^2 dy =
\frac{\pi}{16} \int_0^4 y^2 dy + \pi \int_4^5 (5-y)dy$">|; 

$key = q/displaystyleintfrac{dx}{sqrt{9+x^2}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.gif"
 ALT="$\displaystyle \int \frac{dx}{\sqrt{9 + x^2}} $">|; 

$key = q/Rightarrowdisplaystyleintfrac{2x^2-x+1}{(x+1)(x-1)^2}dx=ln|x+1|+ln|x-1|-frac{1}{x-1}+C,Cin{{mathbb{R}{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="530" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.gif"
 ALT="$\Rightarrow
\displaystyle \int \frac{2x^2 - x + 1}{(x+1)(x-1)^2} dx =
\ln\vert x+1\vert + \ln\vert x-1\vert - \frac{1}{x-1} + C  ,  C \in {\mathbb{R}}$">|; 

$key = q/displaystylearray{{ll}u=3x+1&longrightarrowdu=3dxx=5&longleftrightarrowu=3(5)+1=16x=ell&longleftrightarrowu=3ell+1array{;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.gif"
 ALT="$\displaystyle \begin{array}{ll}
u = 3x + 1 &amp; \longrightarrow du = 3 dx \\\\
x = ...
... u = 3(5)+1 = 16 \\\\
x = \ell &amp; \longleftrightarrow u = 3 \ell + 1 \end{array} $">|; 

$key = q/displaystyleleft{array{{ccc}A+B&=2C-2A&=-1A-B+C&=1array{right.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.gif"
 ALT="$\displaystyle \left\{ \begin{array}{ccc}
A + B &amp;= 2\ C - 2A &amp; = -1 \ A - B +C &amp;= 1
\end{array} \right. $">|; 

$key = q/(x,y)=(0,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.gif"
 ALT="$(x,y) = (0,0)$">|; 

$key = q/displaystyleF(1)=frac{-1}{2}-frac{1}{2}=-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.gif"
 ALT="$\displaystyle F(1) = \frac{-1}{2} - \frac{1}{2} = -1$">|; 

$key = q/displaystyleF(x)=int_{-1}^xf(t)dt,-1leqxleq3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.gif"
 ALT="$\displaystyle F(x) = \int_{-1}^x f(t) dt  ,  -1 \leq x \leq 3$">|; 

$key = q/displaystyleintfrac{dx}{sqrt{9+x^2}}=intfrac{3sec^2(u)du}{sqrt{9+9tan^2(u)}}=intfrac{3sec^2(u)du}{3sec(u)}=intsec(u)du=ln|sec(u)+tan(u)|+C;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="696" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.gif"
 ALT="$\displaystyle \int \frac{dx}{\sqrt{9 + x^2}} = \int \frac{3 \sec^2(u) du}
{\sqr...
...^2(u) du}{3 \sec(u)}
= \int \sec(u) du = \ln \vert \sec(u) + \tan(u) \vert + C$">|; 

1;