%This macro provides all the formulae of page 7
%
%This macro has one parameter
%        1) The width of the math text
\newcommand\TSevenCalculusTwo[1]{%
   \parbox[t]{#1}{%
      \TSevenSeriesCalculusFontSize
      \begin{DisplayFormulae}{15}{\SpaceBeforeFormula}{\TSevenSkipFormulae}{\BigChar}{\StyleBold}
      \Fm{\int \arccos \tfrac{x}{a}\dx = \arccos \tfrac{x}{a} - \sqrt{a^2 - x^2}
          \MathRemark{a > 0}}
      \Fm{\int \arctan \tfrac{x}{a}\dx = x \arctan \tfrac{x}{a} - \tfrac{a}{2} \ln(a^2 + x^2)%
          \MathRemark{a > 0}}
      \Fm{\int \sin^2 (a x)\dx = \tfrac{1}{2a} \big(ax - \sin(ax) \cos(ax)\big)}
      \Fm{\int \cos^2 (a x)\dx = \tfrac{1}{2a} \big(ax + \sin(ax) \cos(ax)\big)}
      \Fm{\int \sec^2 \xdx = \tan x}
      \Fm{\int \csc^2 \xdx = - \cot x}
      \Fm{\int \sin^n \xdx = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n}\int \sin^{n-2} \xdx}
      \Fm{\int \cos^n \xdx = \frac{\cos^{n-1} x \sin x}{n} + \frac{n-1}{n}\int \cos^{n-2} \xdx} 
      \Fm{\int \tan^n \xdx = \frac{\tan^{n-1} x}{n - 1} - \int \tan^{n-2} \xdx
          \MathRemark{n \neq 1}} 
      \Fm{\int \cot^n \xdx = -\frac{\cot^{n-1} x}{n - 1} - \int \cot^{n-2} \xdx
          \MathRemark{n \neq 1}} 
      \Fm{\int \sec^n \xdx = \frac{\tan x\sec^{n-1} x}{n - 1} + \frac{n-2}{n-1}\int \sec^{n-2} \xdx
          \MathRemark{n \neq 1}} 
      \Fm{\int \csc^n \xdx = -\frac{\cot x\csc^{n-1} x}{n - 1} + \frac{n-2}{n-1}\int \csc^{n-2} \xdx
         \MathRemark{n \neq 1}}
      \Fm{\int \sinh \xdx = \cosh x}
      \Fm{\int \cosh \xdx = \sinh x}
      \Fm{\int \tanh \xdx = \ln \vert \cosh x \vert}
      \Fm{\int \coth \xdx = \ln \vert \sinh x \vert}
      \Fm{\int \sech \xdx = \arctan \sinh x }
      \Fm{\int \csch \xdx = \ln\left\vert\tanh \tfrac{x}{2} \right\vert}
      \Fm{\int \sinh^2 \xdx = \tfrac{1}{4} \sinh (2x) - \tfrac{1}{2} x}
      \Fm{\int \cosh^2 \xdx = \tfrac{1}{4} \sinh (2x) + \tfrac{1}{2} x}
      \Fm{\int \sech^2 \xdx = \tanh x}
      \Fm{\int \arcsinh \tfrac{x}{a}\dx = x \arcsinh \tfrac{x}{a} - \sqrt{x^2 + a^2}
          \MathRemark{a > 0}}
      \Fm{\int \arccosh \tfrac{x}{a}\dx = \left\{
          \begin{array}{l}
              \displaystyle 
                  x \arccosh \frac{x}{a} - \sqrt{x^2 + a^2}
                  \MathRemark{\text{if }\arccosh \frac{x}{a} > 0\text{ and }a > 0} \\
              \displaystyle 
                  \rule{0pt}{5ex plus 1ex minus 1ex}%The two line of the array are too tight, this separates them
                  x \arccosh \frac{x}{a} + \sqrt{x^2 + a^2}
                  \MathRemark{\text{if }\arccosh \frac{x}{a} < 0\text{ and }a > 0} \\
          \end{array}
           \right.
      }
      \Fm{\int \arctanh \tfrac{x}{a}\dx = x \arctanh \tfrac{x}{a} +  \tfrac{a}{2} \ln\vert a^2 - x^2\vert}
      \Fm{\int \frac{dx}{\sqrt{a^2 + x^2}}= \ln \left(x + \sqrt{a^2 + x^2}\right)
          \MathRemark{a > 0}}
      \Fm{\int \frac{dx}{a^2 + x^2}= \tfrac{1}{a} \arctan \tfrac{x}{a}
          \MathRemark{a > 0}}
      \Fm{\int \sqrt{a^2 - x^2}\dx = \tfrac{x}{2} \sqrt{a^2 - x^2} + \tfrac{a^2}{2} \arcsin \tfrac{x}{a}
          \MathRemark{a > 0}}
      \Fm{\int (a^2 - x^2)^{3/2}\dx = \tfrac{x}{8} (5a^2 - 2x^2)\sqrt{a^2 - x^2} + \tfrac{3a^4}{8} \arcsin \tfrac{x}{a}
          \MathRemark{a > 0}} 
      \Fm{\int \frac{dx}{\sqrt{a^2 - x^2}} = \arcsin \tfrac{x}{a}
          \MathRemark{a > 0}}
      \Fm{\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln\left\vert\frac{a + x}{a - x}\right\vert}
      \Fm{\int \frac{dx}{(a^2 - x^2)^{3/2}} = \frac{x}{a^2\sqrt{a^2 - x^2}}}
      \Fm{\int \sqrt{a^2 \pm x^2}\dx = \tfrac{x}{2} \sqrt{a^2 \pm x^2} \pm \tfrac{a^2}{2} \ln\left\vert x + \sqrt{a^2 \pm x^2}\right\vert}
      \Fm{\int \frac{dx}{\sqrt{x^2 - a^2}}= \ln\left\vert x + \sqrt{x^2 - a^2}\right\vert
          \MathRemark{a > 0}}
      \Fm{\int \frac{dx}{a x^2 + b x}= \frac{1}{a} \ln\left\vert\frac{x}{a + bx}\right\vert}
      \Fm{\int x \sqrt{a + b x}\dx= \frac{2(3bx - 2a)(a + bx)^{3/2}}{15 b^2}}
      \Fm{\int \frac{\sqrt{a + b x}}{x}\dx= 2\sqrt{a + b x} + a \int \frac{1}{x \sqrt{a + b x}}\dx}
      \Fm{\int \frac{x}{\sqrt{a + b x}}\dx= \frac{1}{\sqrt{2}} \ln\left\vert\frac{\sqrt{a + b x} - \sqrt{a}}{\sqrt{a + b x} + \sqrt{a}}\right\vert
          \MathRemark{a > 0}}
      \Fm{\int \frac{\sqrt{a^2 - x^2}}{x}\dx = \sqrt{a^2 - x^2} - a \ln\left\vert\frac{a + \sqrt{a^2 - x^2}}{x}\right\vert}
      \Fm{\int x \sqrt{a^2 - x^2}\dx = - \tfrac{1}{3} (a^2 - x^2)^{3/2}}
      \Fm{\int x^2 \sqrt{a^2 - x^2}\dx = \tfrac{x}{8} (2x^2 - a^2)\sqrt{a^2 - x^2} + \tfrac{a^4}{8} \arcsin \tfrac{x}{a}
          \MathRemark{a > 0}}
      \Fm{\int \frac{dx}{\sqrt{a^2 - x^2}}= - \tfrac{1}{a} \ln\left\vert\frac{a + \sqrt{a^2 - x^2}}{x}\right\vert}
      \Fm{\int \frac{\xdx}{\sqrt{a^2 - x^2}} = - \sqrt{a^2 - x^2}}
      \Fm{\int \frac{x^2\dx}{\sqrt{a^2 - x^2}} = - \tfrac{x}{2} \sqrt{a^2 - x^2} + \tfrac{a^2}{2} \arcsin \tfrac{x}{a}
          \MathRemark{a > 0}}
      \Fm{\int \frac{\sqrt{a^2 + x^2}}{x}\dx = \sqrt{a^2 + x^2} - a \ln\left\vert\frac{a + \sqrt{a^2 + x^2}}{x}\right\vert}
      \Fm{\int \frac{\sqrt{x^2 - a^2}}{x}\dx = \sqrt{x^2 - a^2} - a \arccos \tfrac{a}{\vert x\vert}
          \MathRemark{a > 0}}
      \Fm{\int x \sqrt{x^2 \pm a^2}\dx = \tfrac{1}{3} (x^2 \pm a^2)^{3/2}}
      \Fm{\int \frac{dx}{x \sqrt{x^2 + a^2}} = \tfrac{1}{a} \ln \left\vert\frac{x}{a + \sqrt{a^2 + x^2}}\right\vert}
      \end{DisplayFormulae}
   }%
}
