(%o0)                                done
(%i1) 
   ****************************************************************************
(%i2)                                 Q1
(%i3) 
   ****************************************************************************
(%i4)                            ***** i *****
                                              2
(%i5)                      integrate(2 x + 3 x , x)
                                     3    2
(%o5)                               x  + x
(%i6)                           ***** ii *****
(%i7)                    integrate(cos(x) + sin(x), x)
(%o7)                           sin(x) - cos(x)
(%i8)                           ***** iii *****
(%i9)                   integrate(exp(- x) + exp(x), x)
                                    x     - x
(%o9)                             %e  - %e
(%i10)                          ***** iv *****
                                                2
(%i11)                  factor(integrate((2 + x) , x))
                                   2
                               x (x  + 6 x + 12)
(%o11)                         -----------------
                                       3
(%i12) 
   ****************************************************************************
(%i13)                                Q2
(%i14) 
   ****************************************************************************
(%i15)                           ***** i *****
(%i16)                      integrate(x sin(x), x)
(%o16)                         sin(x) - x cos(x)
(%i17)                          ***** ii *****
(%i18)                    integrate(exp(x) cos(x), x)
                               x
                             %e  (sin(x) + cos(x))
(%o18)                       ---------------------
                                       2
(%i19)                          ***** iii *****
(%i20)                      integrate(x exp(x), x)
                                            x
(%o20)                            (x - 1) %e
(%i21)                          ***** iv *****
                                      2
(%i22)                     integrate(x  sin(2 x), x)
                                             2
                      4 x sin(2 x) + (2 - 4 x ) cos(2 x)
(%o22)                ----------------------------------
                                      8
(%i23)                           ***** v *****
(%i24)                      integrate(x log(x), x)
                                 2           2
                                x  log(x)   x
(%o24)                          --------- - --
                                    2       4
(%i25)                          ***** vi *****
(%i26)                   integrate(sqrt(x) log(x), x)
                               3/2             3/2
                              x    log(x)   2 x
(%o26)                     2 (----------- - ------)
                                   3          9
(%i27) 
   ****************************************************************************
(%i28)                                Q3
(%i29) 
   ****************************************************************************
(%i30)                           ***** i *****
                                         2     3
(%i31)                     integrate(t (t  - 1) , t)
                                     2     4
                                   (t  - 1)
(%o31)                             ---------
                                       8
(%i32)                          ***** ii *****
                                                4
                                   (3 + sqrt(u))
(%i33)                   integrate(--------------, u)
                                      sqrt(u)
                                              5
                               2 (sqrt(u) + 3)
(%o33)                         ----------------
                                      5
(%i34)                          ***** iii *****
                                              2
(%i35)                    integrate(sqrt(1 - x ), x)
                                                 2
                           asin(x)   x sqrt(1 - x )
(%o35)                     ------- + --------------
                              2            2
(%i36)                          ***** iv *****
                                          2
(%i37)                    integrate(sqrt(x  + 1), x)
                                              2
                           asinh(x)   x sqrt(x  + 1)
(%o37)                     -------- + --------------
                              2             2
(%i38)                           ***** v ****
                                         1
                                      -------
                                      sqrt(x)
(%i39)                   integrate(--------------, x)
                                                2
                                   (sqrt(x) + 1)
                                        2
(%o39)                           - -----------
                                   sqrt(x) + 1
(%i40)                          ***** vi *****
(%i41)               integrate(exp(x) cos(2 + exp(x)), x)
                                       x
(%o41)                           sin(%e  + 2)
(%i42) 
   ****************************************************************************
(%i43)                                Q4
(%i44) 
   ****************************************************************************
(%i45)                           ***** i *****
                                             1
(%i46)             ratsimp(integrate(-----------------, x))
                                     2 cos(x) + sin(x)
                  2 sin(x) + (- sqrt(5) - 1) cos(x) - sqrt(5) - 1
              log(-----------------------------------------------)
                   2 sin(x) + (sqrt(5) - 1) cos(x) + sqrt(5) - 1
(%o46)      - ----------------------------------------------------
                                    sqrt(5)
(%i47)                          ***** ii *****
                                       sin(x)
(%i48)                  integrate(-----------------, x)
                                  2 cos(x) + sin(x)
                      2                                     2
                   sin (x)        sin(x)                 sin (x)
            log(------------- - ---------- - 1)   log(------------- + 1)
                            2   cos(x) + 1                        2
                (cos(x) + 1)                          (cos(x) + 1)
(%o48) 4 (- ----------------------------------- + ----------------------
                            10                              10
                                                                     sin(x)
                                                              atan(----------)
                                                                   cos(x) + 1
                                                            + ----------------)
                                                                     10
(%i49)                          ***** iii *****
                                             1
(%i50)                ratsimp(integrate(------------, x))
                                        2 sin(x) + 1
                   sin(x) + (2 - sqrt(3)) cos(x) - sqrt(3) + 2
               log(-------------------------------------------)
                   sin(x) + (sqrt(3) + 2) cos(x) + sqrt(3) + 2
(%o50)         ------------------------------------------------
                                   sqrt(3)
(%i51) 
   ****************************************************************************
(%i52)                                Q5
(%i53) 
   ****************************************************************************
(%i54)                           ***** i *****
                                         1
(%i55)                    integrate(------------, t)
                                               2
                                    5 + 4 t + t
                                      2 t + 4
(%o55)                           atan(-------)
                                         2
(%i56)                          ***** ii *****
                                         1
(%i57)                   integrate(--------------, t)
                                                2
                                   2 + 3 t + 2 t
                                       4 t + 3
                                2 atan(-------)
                                       sqrt(7)
(%o57)                          ---------------
                                    sqrt(7)
(%i58)                          ***** iii *****
                                        2 t
(%i59)                   integrate(--------------, t)
                                                2
                                   9 + 6 t + 3 t
                                                 2 t + 2
                                            atan(-------)
                             2                     3/2
                        log(t  + 2 t + 3)         2
(%o59)               2 (----------------- - -------------)
                                6             3 sqrt(2)
(%i60) 
   ****************************************************************************
(%i61)                                Q6
(%i62) 
   ****************************************************************************
(%i63)                           ***** i *****
                                             1
(%i64)                 aaa01 : partfrac(------------, x)
                                                   2
                                        2 - 3 x + x
                                   1       1
(%o64)                           ----- - -----
                                 x - 2   x - 1
(%i65)                        integrate(aaa01, x)
(%o65)                      log(x - 2) - log(x - 1)
(%i66)                          ***** ii *****
                                             1
(%i67)               aaa01 : partfrac(---------------, x)
                                               2    3
                                      1 - x - x  + x
                          1           1           1
(%o67)                --------- - --------- + ----------
                      4 (x + 1)   4 (x - 1)            2
                                              2 (x - 1)
(%i68)                        integrate(aaa01, x)
                      log(x + 1)   log(x - 1)       1
(%o68)                ---------- - ---------- - ---------
                          4            4        2 (x - 1)
(%i69)                          ***** iii *****
                                          1 + 2 x
(%i70)                   aaa01 : partfrac(-------, x)
                                           3
                                          x  - 1
                                1         x
(%o70)                        ----- - ----------
                              x - 1    2
                                      x  + x + 1
(%i71)                        integrate(aaa01, x)
                                         2 x + 1
                       2            atan(-------)
                  log(x  + x + 1)        sqrt(3)
(%o71)          - --------------- + ------------- + log(x - 1)
                         2             sqrt(3)
(%i72) 
   ****************************************************************************
(%i73)                                Q7
(%i74) 
   ****************************************************************************
(%i75)                           ***** i *****
                                         x
(%i76)                      integrate(--------, x)
                                             3
                                      (1 + x)
                                    2 x + 1
(%o76)                         - --------------
                                    2
                                 2 x  + 4 x + 2
(%i77)                          ***** ii *****
                                  4 sin(x) - 3 cos(x)
(%i78)                    aaa01 : -------------------
                                   cos(x) + 2 sin(x)
                              4 sin(x) - 3 cos(x)
(%o78)                        -------------------
                               2 sin(x) + cos(x)
                                                              2
                                         2 t             1 - t
                     ev(aaa01, sin(x) = ------, cos(x) = ------) 2
                                         2                2
                                        t  + 1           t  + 1
(%i79)       aaa01 : ---------------------------------------------
                                         2
                                        t  + 1
                                               2
                               8 t     3 (1 - t )
                           2 (------ - ----------)
                               2          2
                              t  + 1     t  + 1
(%o79)                    --------------------------
                                         2
                            2       1 - t     4 t
                          (t  + 1) (------ + ------)
                                     2        2
                                    t  + 1   t  + 1
(%i80)                       aaa01 : factor(aaa01)
                               2 (t + 3) (3 t - 1)
(%o80)                     - -----------------------
                               2        2
                             (t  + 1) (t  - 4 t - 1)
(%i81)                    aaa01 : partfrac(aaa01, t)
                              8 - 4 t      4 t + 2
(%o81)                      ------------ + -------
                             2              2
                            t  - 4 t - 1   t  + 1
(%i82)                    aaa02 : integrate(aaa01, t)
                        2                     2
(%o82)         - 2 log(t  - 4 t - 1) + 2 log(t  + 1) + 2 atan(t)
                                                   x
(%i83)                   aaa02 : ev(aaa01, t = tan(-))
                                                   2
                                    x               x
                          8 - 4 tan(-)        4 tan(-) + 2
                                    2               2
(%o83)               ---------------------- + ------------
                        2 x          x           2 x
                     tan (-) - 4 tan(-) - 1   tan (-) + 1
                          2          2             2
(%i84)                          ratsimp(aaa02)
                                2 x           x
                           6 tan (-) + 16 tan(-) - 6
                                  2           2
(%o84)               - ----------------------------------
                          4 x         3 x          x
                       tan (-) - 4 tan (-) - 4 tan(-) - 1
                            2           2          2
(%i85)                          ***** iii *****
                                                8
                                             cos (x)
(%i86)                   aaa01 : trigexpand(---------)
                                               3
                                            sin (2 x)
                                       5
                                    cos (x)
(%o86)                             ---------
                                        3
                                   8 sin (x)
(%i87)                        integrate(aaa01, x)
                                          2
                                       sin (x)       1
                     - 2 log(sin(x)) + ------- - ---------
                                          2           2
                                                 2 sin (x)
(%o87)               -------------------------------------
                                       8
(%i88)                          ***** iv *****
                                      exp(2 t)
(%i89)                 integrate(------------------, t)
                                 sqrt(exp(2 t) + 1)
                                       2 t
(%o89)                          sqrt(%e    + 1)
(%i90)                           ***** v *****
                                             x
(%i91)            aaa01 : partfrac(---------------------, x)
                                                  2    3
                                   - 4 + 8 x - 5 x  + x
                             1       1        2
(%o91)                     ----- - ----- + --------
                           x - 1   x - 2          2
                                           (x - 2)
(%i92)                        integrate(aaa01, x)
                                                    2
(%o92)                  log(x - 1) - log(x - 2) - -----
                                                  x - 2
(%i93)                          ***** vi *****
                                          2
                                         x
(%i94)                    integrate(------------, x)
                                          3
                                    sqrt(x  - 3)
                                        3
                                2 sqrt(x  - 3)
(%o94)                          --------------
                                      3
(%i95)                          ***** vii *****
                                           2 3/2
(%i96)                     integrate((1 - x )   , x)
                                     2 3/2                 2
                 3 asin(x)   x (1 - x )      3 x sqrt(1 - x )
(%o96)           --------- + ------------- + ----------------
                     8             4                8
(%i97)                         ***** viii *****
                                             5
(%i98)                   integrate(cos(x) sin (x), x)
                                       6
                                    sin (x)
(%o98)                              -------
                                       6
(%i99)                          ***** ix *****
(%i100)              aaa01 : trigreduce(sin(3 x) sin(8 x))
                             cos(5 x)   cos(11 x)
(%o100)                      -------- - ---------
                                2           2
(%i101)                       integrate(aaa01, x)
                             sin(5 x)   sin(11 x)
(%o101)                      -------- - ---------
                                10         22
(%i102) 
   ****************************************************************************
(%i103)                               Q8
(%i104) 
   ****************************************************************************
(%i105)                          ***** i *****
                                         2
(%i106)                         aaa01 : x  - 4
                                     2
(%o106)                             x  - 4
(%i107)  integrate(- aaa01, x, - 2, 1) + integrate(aaa01, x, - 3, - 2)
                                      34
(%o107)                               --
                                      3
(%i108)                          ***** i *****
(%i109)                         aaa01 : log(x)
(%o109)                             log(x)
                                                              1
(%i110)     integrate(aaa01, x, 1, 2) + integrate(- aaa01, x, -, 1)
                                                              2
                                        log(2) + 1
(%o110)                      2 log(2) - ----------
                                            2
(%i111) 
   ****************************************************************************
(%i112)                               Q9
(%i113) 
   ****************************************************************************
                                     2
(%i114)           integrate(sqrt(sinh (x) + 1), x, 0, log(2))
                                       3
(%o114)                                -
                                       4
(%i115) 
   ****************************************************************************
(%i116)                               Q10
(%i117) 
   ****************************************************************************
                                           4
(%i118)                 integrate(pi (x - x ), x, 0, 1)
                                     3 pi
(%o118)                              ----
                                      10
(%i119) 
   ****************************************************************************
(%i120)                               Q11
(%i121) 
   ****************************************************************************
(%i122)                    ********** a ***********
                                             2
(%i123)                area : integrate(1 - x , x, 0, 1)
                                       2
(%o123)                                -
                                       3
                                                4
(%i124)             volume : integrate(pi - pi x , x, 0, 1)
                                     4 pi
(%o124)                              ----
                                      5
(%i125)                    ********** b ***********
                                              2
(%i126)               area : integrate(3 x - x , x, 0, 3)
                                       9
(%o126)                                -
                                       2
                                  2       2
(%i127) volume : - integrate(pi (x  - 2 x) , x, 2, 3)
                               2                            2       2
               + integrate(pi x , x, 1, 3) + integrate(pi (x  - 2 x) , x, 0, 1)
                                     20 pi
(%o127)                              -----
                                       3
(%i128) 
   ****************************************************************************
(%i129)                               Q12
(%i130) 
   ****************************************************************************
(%i131)                    ********** a ***********
(%i132)           area : integrate(exp(1) - exp(x), x, 0, 1)
(%o132)                                1
                        integrate(x (exp(1) - exp(x)), x, 0, 1)
(%i133)         x_bar : ---------------------------------------
                                         area
                                    %e
(%o133)                             -- - 1
                                    2
                          (exp(x) + exp(1)) (exp(1) - exp(x))
                integrate(-----------------------------------, x, 0, 1)
                                           2
(%i134) y_bar : -------------------------------------------------------
                                         area
                                      2
                                    %e    1
                                    --- + -
                                     2    2
(%o134)                             -------
                                       2
(%i135)                         ratsimp(y_bar)
                                      2
                                    %e  + 1
(%o135)                             -------
                                       4
(%i136)                    ********** b ***********
                                             2
(%i137)            area : integrate(- 2 x - x  + 8, x, 0, 2)
                                      28
(%o137)                               --
                                      3
                                              2
                        integrate(x (- 2 x - x  + 8), x, 0, 2)
(%i138)         x_bar : --------------------------------------
                                         area
                                       5
(%o138)                                -
                                       7
                                    2                2
                            (2 x - x  + 8) (- 2 x - x  + 8)
                  integrate(-------------------------------, x, 0, 1)
                                           2
(%i139)   y_bar : ---------------------------------------------------
                                         area
                                      863
(%o139)                               ---
                                      280
(%i140) 
   ****************************************************************************
(%i141)                               Q13
(%i142) 
   ****************************************************************************
(%i143)                     ********** i **********
(%i144) area : integrate(exp(x) - exp(1), x, 1, 2)
                                        + integrate(exp(1) - exp(x), x, - 1, 1)
                           - 1    2          2
(%o144)                  %e    (%e  + 1) + %e  - 2 %e
(%i145)                      area : ratsimp(area)
                               - 1    3     2
(%o145)                      %e    (%e  - %e  + 1)
(%i146)                           float(area)
(%o146)                        5.038653711643048
(%i147)                    ********** ii **********
                        2
(%i148) Iy : integrate(x  (exp(x) - exp(1)), x, 1, 2)
                                                2
                                   + integrate(x  (exp(1) - exp(x)), x, - 1, 1)
                        2            - 1    2
                    6 %e  - 8 %e   %e    (%e  + 15)   4 %e
(%o148)             ------------ + ---------------- - ----
                         3                3            3
(%i149)                        Iy : ratsimp(Iy)
                            - 1      3        2
                          %e    (6 %e  - 11 %e  + 15)
(%o149)                   ---------------------------
                                       3
(%i150)                            float(Iy)
(%o150)                        6.650476032702012
(%i151)                    ********** iii **********
                                              Iy
(%i152)                    ky : sqrt(ratsimp(----))
                                             area
                                    3        2
                           sqrt(6 %e  - 11 %e  + 15)
(%o152)                    -------------------------
                                     3       2
                            sqrt(3 %e  - 3 %e  + 3)
(%i153)                            float(ky)
(%o153)                         1.1488652948528
(%i154)                    ********** iv **********
                                  2
(%i155) Ix : integrate(integrate(y , y, exp(1), exp(x)), x, 1, 2)
                                              2
                       + integrate(integrate(y , y, exp(x), exp(1)), x, - 1, 1)
                      - 3      6          6       3       3
                    %e    (3 %e  + 1)   %e  - 6 %e    4 %e
(%o155)             ----------------- + ----------- + -----
                            9                9          9
(%i156)                        Ix : ratsimp(Ix)
                               - 3    9     6
                             %e    (%e  + %e  + 1)
(%o156)                      ---------------------
                                       9
(%i157)                            float(Ix)
(%o157)                        47.06267972047674
(%i158)                     ********** v **********
                                           Ix
(%i159)                         kx : sqrt(----)
                                          area
                             - 1        9     6
                           %e    sqrt(%e  + %e  + 1)
(%o159)                    -------------------------
                                      3     2
                             3 sqrt(%e  - %e  + 1)
(%i160)                         kx : float(kx)
(%o160)                        3.056195064610983
(%i161)           ******* vi,vii,iix,ix same as before *****
(%i162)                     ********** x **********
                                         2     2
(%i163)                      I0 : sqrt(Iy  + Ix )
                 - 6    9     6     2     - 2      3        2      2
               %e    (%e  + %e  + 1)    %e    (6 %e  - 11 %e  + 15)
(%o163)   sqrt(---------------------- + ----------------------------)
                         81                          9
(%i164)                            float(I0)
(%o164)                        47.53024988292946
(%i165)                     ********* xi **********
                                           I0
(%i166)                         k0 : sqrt(----)
                                          area
                    - 6    9     6     2     - 2      3        2      2
                  %e    (%e  + %e  + 1)    %e    (6 %e  - 11 %e  + 15)  1/4
        sqrt(%e) (---------------------- + ----------------------------)
                            81                          9
(%o166) -------------------------------------------------------------------
                                       3     2
                                sqrt(%e  - %e  + 1)
(%i167)                            float(k0)
(%o167)                        3.071339271146625
(%o167)                          answers02.max