2017年9月16日土曜日

学習環境

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第20章(面積、体積、長さ - 積分法の応用)、20.1(面積)、面積の公式、問1.を取り組んでみる。


    1. y=(x+1)(x1)11(0(x21))dx=210x2dx+2101dx=2[13x3]10+2[x]10=23+2=43

    2. y=(1x)(x2+x+1)10(1x3)dx=[x14x4]10=114=34

    3. y=x2(1x)10(x2x3)dx=[13x314x4]10=1314=112

    4. y=(x22x3)=(x3)(x+1)31(x2+2x+3)dx=[13x3+x2+3x]31=(9+9+9)(13+13)=9+53=323

    5. x2=x2+3x+52x23x5=0(x+1)(2x5)=0x=1,5220((x2+3x+5)x2)dx=20(2x2+3x+5)dx=[23x3+32x2+5x]20=163+6+10=323

    6. y=xx2=xx4=xx(x31)=0x=0,110(xx2)dx=[23x3213x3]10=2313=13

    7. y2=3xx=y234yy2=y2312y3y2=y24y212y=04y(y3)=0y=0,330((4yy2)y23)dy=30(4y43y2)dy=[2y249y3]30=1812=6

    8. π40cosxdx=[sinx]π40=sinπ4sin0=12

    9. sinx=cosxx=π4π40(cosxsinx)dx=[sinx+cosx]π40=(sinπ4+cosπ4)(sin0+cos0)=12+12(0+1)=221=21

    10. =π0sinxdx2ππsinxdx+3π2πsinxdx=2π0sinxdx2ππsinxdx=2[cosx]π0[cosx]2ππ=2(cosπ+cos0)+(cos2πcosπ)=2(1+1)+(1+1)=6

    11. y=1xy=52x52x=1x52xx21=02x25x+2=0x=5±25164=5±94=5±34=12,2212((52x)1x)dx=212(52x1x)dx=[52x12x2logx]212=(52log2)(5418log12)=(3log2)(98log12)=158log2+log12=158log2log2=1582log2

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Integral, sqrt, cos, pi, sin, Rational

print('1.')
x = symbols('x')
fs = [(0 - (x ** 2 - 1), (-1, 1)),
      (1 - x ** 3, (0, 1)),
      (x ** 2 - x ** 3, (0, 1)),
      (- x ** 2 + 2 * x + 3, (-1, 3)),
      (-x ** 2 + 3 * x + 5 - x ** 2, (0, 2)),
      (sqrt(x) - x ** 2, (0, 1)),
      ((4 * x - x ** 2) - x ** Rational(2, 3), (0, 3)),
      (cos(x), (0, pi / 4)),
      (cos(x) - sin(x), (0, pi / 4)),
      (abs(sin(x)), (0, 3 * pi)),
      (Rational(5, 2) - x - 1 / x, (Rational(1, 2), 2))]

for i, (f, (x1, x2)) in enumerate(fs, 1):
    print(f'({i})')
    I = Integral(f, (x, x1, x2))
    for g in [I, I.doit()]:
        pprint(g)
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
1.
(1)
1               
⌠               
⎮  ⎛   2    ⎞   
⎮  ⎝- x  + 1⎠ dx
⌡               
-1              

4/3


(2)
1              
⌠              
⎮ ⎛   3    ⎞   
⎮ ⎝- x  + 1⎠ dx
⌡              
0              

3/4


(3)
1               
⌠               
⎮ ⎛   3    2⎞   
⎮ ⎝- x  + x ⎠ dx
⌡               
0               

1/12


(4)
3                     
⌠                     
⎮  ⎛   2          ⎞   
⎮  ⎝- x  + 2⋅x + 3⎠ dx
⌡                     
-1                    

32/3


(5)
2                      
⌠                      
⎮ ⎛     2          ⎞   
⎮ ⎝- 2⋅x  + 3⋅x + 5⎠ dx
⌡                      
0                      

32/3


(6)
1             
⌠             
⎮ ⎛      2⎞   
⎮ ⎝√x - x ⎠ dx
⌡             
0             

1/3


(7)
3                       
⌠                       
⎮ ⎛   2/3    2      ⎞   
⎮ ⎝- x    - x  + 4⋅x⎠ dx
⌡                       
0                       

     2/3    
  9⋅3       
- ────── + 9
    5       


(8)
π          
─          
4          
⌠          
⎮ cos(x) dx
⌡          
0          

√2
──
2 


(9)
π                      
─                      
4                      
⌠                      
⎮ (-sin(x) + cos(x)) dx
⌡                      
0                      

-1 + √2


(10)
3⋅π            
 ⌠             
 ⎮  │sin(x)│ dx
 ⌡             
 0             

3⋅π            
 ⌠             
 ⎮  │sin(x)│ dx
 ⌡             
 0             


(11)
 2                 
 ⌠                 
 ⎮  ⎛     5   1⎞   
 ⎮  ⎜-x + ─ - ─⎟ dx
 ⎮  ⎝     2   x⎠   
 ⌡                 
1/2                

-2⋅log(2) + 15/8


$

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