20/07/17 13:11:56.62 7KjVzawt.net
>>234
(与式) = [ -exp(-x)arctan(x) ](0,∞) + ∫[0,∞] exp(-x)/(1+xx) dx
= ∫[0,∞] exp(-x)/(1+xx) dx
= f(1)
= 0.6214496242358
ここに
f(a) = ∫[0,∞] exp(-ax)/(1+xx) dx, (a>0)
より
f(a) + f "(a) = ∫[0,∞] exp(-ax) dx = 1/a,
これを解くと
f(a) = ∫[0,∞] sin(y)/(y+a) dy
= ∫[a,∞] sin(θ-a)/θ dθ
= ∫[a,∞] {-cosθ･sin(a) + sinθ･cos(a)}/θ dθ
= Ci(a)sin(a) + {π/2 - Si(a)}cos(a),
ここに
Ci(a) = -∫[a,∞] (cosθ)/θ dθ, 余弦積分
Si(a) = ∫[0,a] (sinθ)/θ dθ, 正弦積分
これに
Ci(1) = 0.337403922901
π/2 - Si(1) = 0.6247132564277
cos(1) = 0.54030230586814
sin(1) = 0.8414709848079
を入れる。