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) ≒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 ) |
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≦t≦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)≒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 |
初版: 2007.6.20
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