求解定积分;(ax+b)/(cx+d),从0.1到0.2,其中a=cos(0.1),b=[sin(0.1)-0.1*cos(0.1)],c=0.2,d=0.99;

Question

求解定积分; (ax+b)/(cx+d),从0.1 到0.2,其中a=cos(0.1),b=[sin(0.1)-0.1*cos(0.1)],c=0.2,d=0.99;

Answer 1

∫ (ax + b)/(cx + d) dx,先求不定积分比较方便.
= ∫ [(a/c)(cx + d - d) + b]/(cx + d) dx
= (a/c) ∫ dx + (b - ad/c)∫ dx/(cx + d)
= (a/c) • x + (b - ad/c)(1/c)∫ d(cx + d)/(cx + d)
= (a/c) • x + (b/c - ad/c²)ln(cx + d) |(0.0.2),求定积分就代入上下限
代入a = cos(0.1),b = sin(0.1) - (0.1)cos(0.1),c = 0.2,d = 0.99
原式 = 0.0146496...