Instructor Hentzel
Office Phone: 515-294-8141
E-mail:   hentzel@iastate.edu
Math Department Fax:      515-294-5454
http://www.math.iastate.edu/hentzel/class.166.08

Textbook: Calculus by Varberg, Purcell, Rigdon, ninth edition. 

Wednesday, February 6, 7.1 Int. Rules    

p386: 26,36,42

Main Idea:  Review of integration. 

Key Words: Completing the square, Substitution 

Goal: Learn to recognize the integral pieces. 
--------------------------------------------------------
Previous assignment:         p324:(4),(12),(20),(22)
        
Page 324 Problem  4 

Find the following antiderivatives

                 1
          INT  ------- dx
                0.99
               x


               0.01 
Solution:     x
           -------  + C
             0.01
................................
Page 324 Problem 12 

Evaluate the expressions at the given values.

       n/2 
(1+2/n)   ; n = 1,10,100,1000


f[n_] := (1+2/n)^(n/2);

f[{1,10,100,1000}]

{1.73205, 2.48832, 2.69159, 2.71557}

-------------------------------------------
Get["font.math"];
f[n_] := (1.0+2.0/n)^(n/2.0);
A = Table[{n,f[n]},{n,1,2000,50}];
P1 = ListPlot[A,PlotStyle->{RGBColor[1,0,0],PointSize[0.01]}];
P2 = Plot[E,{x,0,2000},PlotStyle->{RGBColor[0,0,1]}];
P3 = Show[P1,P2,PlotLabel->"P324 P12 f[n]=(1+2/n)^(n/2)",
     PlotRange->All];
Display["p12.ps",P3];
-------------------------------------------

Page 324 Problem 20 

Find all of the following in terms of x,

Sin[theta], Cos[theta], Tan[theta], Ctn[theta], Sec[theta], Csc[theta];






                         /|
                        / |
                       /  |
                      /   |          2
                   1 /    |  Sqrt[1-x ]
                    /     |
                   /      |
                  /theta  |
                 /________|
                      x

                     2
Sin[theta] = Sqrt[1-x  ]


Cos[theta] := x
                      2
Tan[theta] := Sqrt[1-x ]/x

                        2
Ctn[theta] := x/Sqrt[1-x ]


Sec[theta] := 1/x

                        2
Csc[theta] := 1/Sqrt[1-x  ]

----------------------------------------------------------
Page 324 Problem 22 

Solve the differential equation subject to the given condition.

 /    Cos[x]
y  = --------   Through (0,4)
         y

   /
y y  = Cos[x]

     2
1/2 y  = Sin[x] + C

    y = Sqrt[2 Sin[x] + C]

  /   /
\/   y  = 1/2 (2 Sin[x] + C)^(-1/2) 2 Cos[x]
           
                Cos[x]            Cos[x]
        = ------------------- =  ---------
           Sqrt[2 Sin[x] +C]        y

4 = Sqrt[C]  so C = 16


     y = Sqrt[ 2 Sin[x] + 16]













......................................
Get["font.math"];
u[x_,y_] := 1/Sqrt[x^2+y^2] {x,y};
f[x_] := Sqrt[2 Sin[x] + 16];
P1 = Plot[f[x],{x,-2 Pi,2 Pi},PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}];
d = 2 Pi/10;
A = Table[ ParametricPlot[ {i*d,j*d}+ t u[j*d,Cos[i*d]],{t,-1/4,1/4},
       PlotStyle->{Thickness[0.005]}], {i,-10,10},{j,-10,10}];
B = Table[{i*d,j*d},{i,-10,10},{j,-10,10}];
B = Flatten[B,1];
P2 = Show[A];
P3 = ListPlot[B,PlotStyle->{RGBColor[1,0,0],PointSize[0.005]}];
P4 = Show[P1,P2,P3,PlotLabel->"P324 p22: y' = Cos[x]/y @ (0,4)",
     PlotRange->All];
Display["p22x.ps",P4];
g[x_,c_] := Sqrt[2 Sin[x] + c];
H = Table[ Plot[g[x,c],{x,-2 Pi,2 Pi},
    PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}],{c,-2, 2,0.5}];
K = Table[ Plot[g[x,c],{x,-2 Pi,2 Pi},
    PlotStyle->{RGBColor[1,0,0],Thickness[0.005]}],{c, 2,36,2.0}];
P5 = Show[P1,P2,P3,H,K,PlotLabel->"P324 p22: y' = Cos[x]/y @ (0,4)",
     PlotRange->All];
Display["p22y.ps",P5];
................................................. 

Example 1 page 384

                   x
            INT------------- dx
                     2  2
                Cos[x  ]  


                 2  2
ans  =  1/2 Tan[x  ]   + C    


Example 2    

                         3
            INT    ------------ dx
                            2
                   Sqrt[5-9x  ]


                3x
ans = ArcSin(---------) + C
              Sqrt[5]

Example 3   

                       1/x
                    6 e
             INT   ------  dx
                     2
                    x  
           1/x
ans =  -6 e    + C



Example 4                 x 
                         e  
               INT  ---------------  dx
                           2x
                      4+9 e




                      x 
                   3 e 
          ArcTan[ -----]    
                    2
ans  =    --------------  + C
                 6

---------------------------------------------------------
Example 5   
                           2
               INT  x Cos[x ]  dx


                 2
ans =   1/2 Sin[x ] + C
--------------------------------------------------------
Example 6  
                         Tan[t] 
                        a
                INT  ----------  dt
                           2
                        Cos [t]  

             Tan[t]
            a
ans =  ----------
          ln(a)
------------------------------------------------------
Example 7

            t=5          2
            INT  t Sqrt[t  - 4] dt
            t=2

        _    2    3/2   _  t=5
       |   (t - 4)       |                3/2
ans =  | ---------       |        = 1/3 21
       |_   3/2 2       _| t=2
-------------------------------------------------------
Example 8
                x=3  3        4
                INT x   Sqrt[x  + 11] dx
                x=1

        _             _
       |    4    3/2   | x=3
       |  (x +11)      |                 3/2   3/2
ans =  |  --------     |        = 1/6 (92  - 12   )
       |_    6        _| x=1
------------------------------------------------------
Example  
                          7
               INT ---------------- dx
                       2
                      x -6x+25

                     x-3
ans =  7/4 ArcTan[ ------- ] + C
                      4
-------------------------------------------------------
Example  
                       2
                      x - x
                 INT ------- dx
                      x + 1
         2
        x
ans = ----- - 2x + 2 ln|x+1| + C
        2