定積分

著者:Mathcot.H.I.
 
初版: 2007.6.20
Update:2013.03.06


定積分

f(t),[積分範囲下限,上限
積分値
x, [a,b]
x2, [a,b]
x2, [-1,1]
1-x2, [0,1]
x(x2-a2) , [-a,0]
√x, [0,a] (a>0)
√x, [0,1]
(b2 -a2)/2
(b3 -a3)/3
 2/3
 4/3
(1/4)a2
(2/3)a3/2
 2/3
√(1-x2), [0,1]: 4分割円面積
√(1-x2), [0,1/√2]: 分割円面積
√(1-x2), [1/√2, 1]: 分割円面積
b√(1-(x/a)2), [0,a]:4分割楕円面積
π/4 
(π+2)/8 
(π-2)/8 
 abπ/4 
√((1-x2)/(x-x2)) , [0, 1]
√2+ln(1+√2)≒2.29559
放物線長 1-x2
√(1+4t2) , [-1,1]
√5 + (1/2)sinh-1(2)=   
√5 + (1/2)ln(2+√5)=2.957885716
π(xR/h)2, [0,h]:円錐体積(高さh,底辺半径R)  hπR2/3
e-x, [0,1]
xe-x, [0,1]
x2e-x, [0,1]
x3e-x, [0,1]
1/√(ex-1),[0,∞]
 1-(1/e)
 
 1-(2/e)
 2-(5/e)

 6-(16/e)
π
(ex-1)/x ,  [0,1]
Ei(1)-γ≒1.317902151454403894860008844249
exp(-t2), [0,x]
(1/2)(√π)erf(x)
1/(1+x2), [0,1]
1/(1+x2), [0,∞]
π/4

π/2
1/(1+x3), [1,∞]
=(1/3){1/(x+1)-(x-2)/(x2-x+1)}
[(1/6)ln((x+1)2/(x2-x+1))+tan-1((2x-1)/√3)/√3] [1,∞]
=π/(3√3) -(1/3)ln(2)
1/(x3-1), [-∞,0]
=(1/3){1/(x-1)-(x+2)/(x2+x+1)}
[(1/6)ln((x-1)2/(x2+x+1))-tan-1((2x+1)/√3)/√3] [-∞,0]
=-2π/(3√3)
1/(1+x4), [-∞,∞]
x2/(1+x4), [-∞,∞]
x/(1+x4), [0,∞]
π/√2  
π/√2  
π/4  
1/(1+x6), [-∞,∞]
x2/(1+x6), [-∞,∞]
x4/(1+x6), [-∞,∞]
x/(1+x6), [0,∞]
x3/(1+x6), [0,∞]
2π/3  
π/3  
2π/3  
√3π/9  
√3π/9  
1/(1+x8), [-∞,∞]
x2/(1+x8), [-∞,∞]
x4/(1+x8), [-∞,∞]
x6/(1+x8), [-∞,∞]
x/(1+x8), [0,∞]
x3/(1+x8), [0,∞]
x5/(1+x8), [0,∞]
π/{4sin(π/8)}

π/{4sin(3π/8)}

π/{4sin(3π/8)}
π/{4sin(π/8)}

√2π/8

π/8
√2π/8
1/((k-x2)2+a2x2) , [-∞,∞] (k>0,x>0)
1/((1-x2)2+x2)=1/(x4-x2+1) , [-∞,∞]
1/((1-x2)2+4x2)=1/(x2+1)2 , [-∞,∞]
1/((2-x2)2+4x2)=1/(x4+4)2 , [-∞,∞]
1/((3-x2)2+4x2)=1/(x4-2x2+9) , [-∞,∞]
1/((1-x2)2+9x2)=1/(x4+7x2+1) , [-∞,∞]
1/((2-x2)2+9x2)=1/(x4+5x2+4) , [-∞,∞]
π/(ak)    
π 
π/2  
π/4  
π/6  
π/3  
π/6  
1/((2+x2)2+x2) , [-∞,∞]
1/((8+x2)2+4x2) , [-∞,∞]
π/6  
π/48   
ln x, [1,a] (a>1)
ln x, [1,2]
ln x, [a,b] (0<a<b)
 a(ln a)-a+1
 2(ln 2)-1
 b(ln b)-a(ln a)-b+a
三角関数 を含む定積分

sin(x) , [0,π]
sin(x) , cos(x) , [0,π/2]
tan(x) , [0,π/4]
 2
 1
(ln2)/2
sin2(x) , cos2(x)  [0,π/2]
sin3(x) , cos3(x)  [0,π/2]
sin4(x) , cos4(x)  [0,π/2]
sin5(x) , cos5(x)  [0,π/2]
π/4 
2/3  
3π/16 
8/15  
x cos x , [0,π/2]
 (π/2)-1
1/{1±(1/2)cos x }, [-π,π]
1/{1±(1/3)cos x }, [-π,π]
1/{1±(1/4)cos x }, [-π,π]
1/{1±(1/5)cos x }, [-π,π]
1/{1±(1/6)cos x }, [-π,π]
1/{1±(1/7)cos x }, [-π,π]
1/{1±(1/8)cos x }, [-π,π]
1/{1±(1/9)cos x }, [-π,π]
1/{1±(2/3)cos x }, [-π,π]
1/{1±(3/4)cos x }, [-π,π]
1/{1±(2/5)cos x }, [-π,π]
1/{1±(3/5)cos x }, [-π,π]
1/{1±(4/5)cos x }, [-π,π]
1/{1±(5/6)cos x }, [-π,π]
1/{1±(2/7)cos x }, [-π,π]
1/{1±(3/7)cos x }, [-π,π]
1/{1±(4/7)cos x }, [-π,π]
1/{1±(5/7)cos x }, [-π,π]
1/{1±(6/7)cos x }, [-π,π]
1/{1±(3/8)cos x }, [-π,π]
1/{1±(5/8)cos x }, [-π,π]
1/{1±(7/8)cos x }, [-π,π]
1/{1±(2/9)cos x }, [-π,π]
1/{1±(4/9)cos x }, [-π,π]
1/{1±(5/9)cos x }, [-π,π]
1/{1±(7/9)cos x }, [-π,π]
4π/√3
3π/√2
8π/√15
5π/√6
12π/√35
7π/(2√3)
16π/(3√7)
16π/(2√5)
6π/√5
8π/√7
10π/√21
5π/2
10π/3
12π/√11
14π/(3√5)
7π/√10
14π/√33
7π/√6
14π/√13
16π/√55
16π/√39
16π/√15
18π/√77
18π/√65
9π/√14
9π/(2√2)
1/{1±k*cos x) ,k^2<1, [0,π]

1/[1+(cos x)2], [0,2π]
π√2
cos(2x)/{1+(cos(2x))^2}, [0,π/2]
0
sin3(x)/(sin(x)+cos(x)) , [0,π/2]
cos3(x)/(sin(x)+cos(x)) , [0,π/2]
(π-1)/4

(π-1)/4
sin(x)/x , [0,∞]
sin2(x)/x2 , [0,∞]
sin3(x)/x3 , [0,∞]
sin4(x)/x4 , [0,∞]
sin5(x)/x5 , [0,∞]
sin6(x)/x6 , [0,∞]
sin7(x)/x7 , [0,∞]
sin8(x)/x8 , [0,∞]
sin9(x)/x9 , [0,∞]
sin10(x)/x10 , [0,∞]
π/2
π/2 
3π/8 
π/3 
115π/384 
11π/40 
5887π/23040 
151π/630 
259723π/1146880 
15619π/72576 
sin(x)/x , [-∞,∞]
sin2(x)/x2 , [-∞,∞]
sin3(x)/x3 , [-∞,∞]
sin4(x)/x4 ,[-∞,∞]
sin5(x)/x5 , [[-∞,∞]
sin6(x)/x6 , [-∞,∞]
sin7(x)/x7 , [-∞,∞]
sin8(x)/x8 , [-∞,∞]
sin9(x)/x9 , [-∞,∞]
sin10(x)/x10 , [-∞,∞]
(フーリエ積分公式で利用して手計算可能)
π
π 
3π/4 
2π/3 
115π/192 
11π/20 
5887π/11520 
151π/315 
259723π/573440 
15619π/36288 
sin(x)cos(x)/x , [0,∞]
sin(x)cos(ax)/x, [0,∞] (0<=a<1)
sin(x)cos(ax)/x, [0,∞] (a>1)
π/4
π/2
 0
sin(πx)/(x(1-x2)) , [0, ∞]
π
x sin(x)/(x2+a2), [0,∞]
πe-a/2 (a>0)
tan-1(x) , [0, 1]
(π/4)-{(ln2)/2}
cosh(x), [-1,1]
1/cosh(x), [-1,1]
1/cosh(x), [-∞,∞]
 e-(1/e)
 2tan-1(e)-2tan-1(1/e)
 π
sin(x)の曲線長
√(1+cos2 x), [0, π]

 2√2 EllipticE(1/2, √2) &fallingdotseq;3.820197788
r=1-cosθ, [0, 2π]の曲線長
2√2 √(1-cosθ), [0,π]

8
 exp(-a cos(x)) cos(sin(x)), [0,π/2]
 exp(-cos(x)) cos(sin(x)), [0,π/2]
= -Si(a) (Si(x):正弦積分)
= ∫(1,∞) sin(x)/x dx=-Si(1)
exp(-4/(3+(√5)sin(2x))),[0,π]
integrate(exp(-4/(3+5^(1/2)*sin(2*x))),x,0,Pi)
=0.767139...
ガ ウス積分 ∫[-∞,∞] exp(-ax2)dx, (a>0)
 exp(-ax2), [-∞,∞] (a>0)
 exp(-x2), [-∞,∞]
 √(π/a)

 √π
 exp(-x2), [0,∞]
 x exp(-ax2), [0,∞]
 x2exp(-ax2), [0,∞]
 x3exp(-ax2), [0,∞]
 x4 exp(-x2), [0,∞]
 (√π)/2

  1/(2a)
 (√π)/(4a√a)

 1/(2a2)

(3/8)√π
∫[0,∞] (x4)exp(-x2)dt の導出 ∫[0,∞]  x4exp(-x2) dx ← x^=uで置換,dx=du/(2√u)
=(1/2)∫[0,∞] u3/2 e-u du
=(1/2) Γ(5/2)
=(1/2) (3/2) (1/2) Γ(1/2)
=(3/8)√π
 {1/(1+x)}-ln{1+(1/x)} [1,∞]
[Maple] h:=∫(1/(1+x) -ln(1+ 1/x))dx;
  h:=-x* ln((1+x)/x);
 h1:=limit(x*ln(x/(1+x)),x=∞);
  h1:=-1
 h1-h|x=1;
  ln(2) -1
1/(x2+a2)3, [0,∞]
 3π/(16a5)
x+1-2x, [0,1]
 (3/2)- ln 2
x/√(4x-x2) , [1,4]
√3 +4π/3
オイラー 積分

第一種オイラー積分(β関数)B(x,y)
=∫[0,1] tx-1(1-t)y-1dt=Γ(x)Γ(y)/Γ(x+y)
第二種オイラー積分(Γ関数)Γ(z)
=∫[0,∞]tz-1 e-t dt (Re z>0)
=lim[n→∞] nzn!/Πk=0n (z+k)
 e-t t-1/2, [0,∞]
 Γ(1/2)
Γ(n)
Γ(x+1)=xΓ(x)
=(n-1)!

B(n,m)=(n-1)!(m-1)!/(n+m-1)! =(n+m)/( nm m+nCn )
参考URL
[1]
[2]
[3]ガ ウス積分Wikipedia
[4]ガウ ス積 分2
[5]オ イラー積分Wikipedia
[6]ガ ンマΓ関数Wikipedia
[7]ハンケルの積分表示Wikipedia
[8]

媒介変数 による定積分
x=f(t), y=g(t)
[演習1] 媒介変数t (0&LessFullEqual;t&LessFullEqual;2π)で
与えられる点(x,y)がつくる曲線αで
囲まれた領域の面積Sを求めよ。
x=cos(2t)
y=t sin(t)



[解答]
dx=-2sin(2t)dt
∫ydx=∫t sin(t)(-2)sin(2t)dt=∫2{cos(3t)-cos(t)}dt
曲線を
(1)t=0〜π/2
(2)t=π/2〜π
(3)t=π〜3π/2
(4)t=3π/2〜2π
の部分に分けて考える。それぞれの区間でxとtを対応させると
(1) x=-1〜1の時 t=π/2〜0で
曲線とX軸とxの範囲で囲まれる面積
S1=∫[π/2,0]ydx=(2π/3)-(8/9)&fallingdotseq;1.05506
(2)x=-1〜1の時 t=π/2〜πで
曲線とX軸とxの範囲で囲まれる面積
S2=∫[π/2,π]ydx=(2π/3)+(8/9)
S12=S2-S1=16/9
(3)x=-1〜1の時 t=π〜3π/2で
曲線とX軸とxの範囲で囲まれる面積
S3=∫[π,3π/2](-y)dx=2π-(8/9)
(4)x=-1〜1の時 t=3π/2〜2πで
S4=∫[3π/2,π](-y)dx=2π+(8/9)
S34=S4-S3=16/9
2つの曲線領域の合計面積は
S=S12+S34=32/9

参考URL
[1]基 本的な積分公式
[2]公 式(積分)(東海大学)
[3] 不定積分サイト:integrals.wolfram.com/index.jsp
[4] Common Integrals
[5] COMMON SUBSTITUTIONS
[6] Common Integrals INTEGRALS CONTAINING ax+b
[7] INTEGRALS CONTAINING THE SQUARE ROOT OF ax+b
[8] INTEGRALS CONTAINING ax+b AND px+q
[9] Tables
[10]
[11]
[12]
[13]

Copyright(C) 2007-2013 Mathcot.H.I. All rights reserved.

参考URL:
初版: 2007.6.20
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inserted by FC2 system inserted
      by FC2 system inserted by FC2 system