(%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