2020年1月15日水曜日

学習環境

解析入門(上) (松坂和夫 数学入門シリーズ 4) (松坂 和夫(著)、岩波書店)の第8章(積分の計算)、8.2(定積分の計算)、問題6の解答を求めてみる。



    1. t=x+x2-1

      とおくと、

      dtdx=1+xx2-1=x+x2-1x2-11dxx+ax2-1=11x+ax2-1·x2-1x+x2-1dt=1dtx+ax+x2-1=1dttx+at-x=x2-1x2-2tx+t2=x2-1x=t2+12t1dttx+a=1dttt2+12t+a=12t2+2at+1dt=21dtt+a2+1-a2s=t+adsdt=121dtt+a2+1-a2=2adss2+1-a2=2limbabdss2+1-a2

      場合分け。

      a<1

      の場合、

      2limbabdss2+1-a2=2limnabdss2+1-a22=2limb11-a2arctans1-a2n=21-a2limbarctanb1-a2-arctana1-a2=21-a2π2-arctana1-a2

      また、

      a=1

      の場合、

      2limbabdss2=2limb-1sab=2limb-1b+1a=2a

      また、

      a>1

      の場合、

      2limbabdss2+1-a2=2limbabdss2-a2-12As+a2-1+Bs-a2-1=A+Bs+B-Aa2-1s2-a2-12A+B=0B-A=1a2-12B=1a2-1B=12a2-1A=-12a2-12limbabdss2+1-a2=1a2-1limbab1s-a2-1-1s+a2-1ds=1a2-1limblogs-a2-1s+a2-1ab=1a2-1limblogb-a2-1b+a2-1-loga-a2-1a+a2-1=1a2-1loga+a2-1a-a2-1

    2. t=1+2acosθ+a2dtdθ=-2asinθ0πsinθ1+2acosθ+a2dθ=-12aa+2a+1a2-2a+11tdt=-12a·2ta2+2a+1a2-2a+1=-1aa2-2a+1-a2+2a+1=-1aa-12+a+12a1-1aa-1+a+1=-2a1-1a1-a+a+1=-2aa-1-1a1-a-a-1=2

コード

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, sqrt, sin, cos, oo, pi, plot

print('6.')

x, a = symbols('x, a', real=True)
f = 1 / ((x + a) * sqrt(x ** 2 - 1))
g = sin(x) / sqrt(1 + 2 * a * cos(x) + a ** 2)
xs = [(1, oo), (0, pi)]
for i, (h, (x1, x2)) in enumerate(zip([f, g], xs), 1):
    print(f'({i})')
    I = Integral(h, (x, x1, x2))
    for o in [I, I.doit()]:
        pprint(o.simplify())
        print()


p = plot(*[h.subs({a: a0})
           for h in [f, g]
           for a0 in range(-2, 3)],
         (x, -5, 5),
         ylim=(-5, 5),
         legend=False,
         show=False)

colors = ['red', 'green', 'blue', 'brown', 'orange',
          'purple', 'pink', 'gray', 'skyblue', 'yellow']

for o, color in zip(p, colors):
    o.line_color = color

for o in zip(p, colors):
    pprint(o)
    print()

p.show()
p.save('sample6.png')

入出力結果(Zsh、PowerShell、Terminal、Jupyter(IPython))

% ./sample6.py 
6.
(1)
∞                       
⌠                       
⎮          1            
⎮ ─────────────────── dx
⎮            ________   
⎮           ╱  2        
⎮ (a + x)⋅╲╱  x  - 1    
⌡                       
1                       

⎧⎧a⋅(2⋅acosh(│a│) - ⅈ⋅π) + ⅈ⋅π⋅│a│       2                          
⎪⎪────────────────────────────────  for a  > 1                      
⎪⎪            ________                                              
⎪⎪           ╱  2                                                   
⎪⎪       2⋅╲╱  a  - 1 ⋅│a│                                          
⎪⎪                                                                  
⎪⎨                     π⋅│a│                    for 2⋅│arg(a)│ < 2⋅π
⎪⎪      -a⋅asin(│a│) + ─────                                        
⎪⎪                       2                                          
⎪⎪      ────────────────────        otherwise                       
⎪⎪           ________                                               
⎨⎪          ╱      2                                                
⎪⎩        ╲╱  1 - a  ⋅│a│                                           
⎪                                                                   
⎪          ∞                                                        
⎪          ⌠                                                        
⎪          ⎮          1                                             
⎪          ⎮ ─────────────────── dx                  otherwise      
⎪          ⎮            ________                                    
⎪          ⎮           ╱  2                                         
⎪          ⎮ (a + x)⋅╲╱  x  - 1                                     
⎪          ⌡                                                        
⎩          1                                                        

(2)
π                            
⌠                            
⎮          sin(x)            
⎮ ──────────────────────── dx
⎮    _____________________   
⎮   ╱  2                     
⎮ ╲╱  a  + 2⋅a⋅cos(x) + 1    
⌡                            
0                            

⎧     ______________      ______________           
⎪    ╱  2                ╱  2                      
⎪- ╲╱  a  - 2⋅a + 1  + ╲╱  a  + 2⋅a + 1            
⎨───────────────────────────────────────  for a ≠ 0
⎪                   a                              
⎪                                                  
⎩                   2                     otherwise

(cartesian line: 1/((x - 2)*sqrt(x**2 - 1)) for x over (-5.0, 5.0), red)

(cartesian line: 1/((x - 1)*sqrt(x**2 - 1)) for x over (-5.0, 5.0), green)

(cartesian line: 1/(x*sqrt(x**2 - 1)) for x over (-5.0, 5.0), blue)

(cartesian line: 1/((x + 1)*sqrt(x**2 - 1)) for x over (-5.0, 5.0), brown)

(cartesian line: 1/((x + 2)*sqrt(x**2 - 1)) for x over (-5.0, 5.0), orange)

(cartesian line: sin(x)/sqrt(5 - 4*cos(x)) for x over (-5.0, 5.0), purple)

(cartesian line: sin(x)/sqrt(2 - 2*cos(x)) for x over (-5.0, 5.0), pink)

(cartesian line: sin(x) for x over (-5.0, 5.0), gray)

(cartesian line: sin(x)/sqrt(2*cos(x) + 2) for x over (-5.0, 5.0), skyblue)

(cartesian line: sin(x)/sqrt(4*cos(x) + 5) for x over (-5.0, 5.0), yellow)

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