Question

高数上册习题5-4定积分第1题(5)

Answer 1

=(t->∞) limit [(-p exp(-pt) sin(wt)-w exp(-pt) cos(wt))/(p²+w²)] - [-p exp(-p*0) sin(w*0)-w exp(-p*0) cos(w*0))/(p²+w²)]
=0+[p exp(0) sin(0)+w exp(0) cos(0))/(p²+w²)
=w/(p²+w²)

追问

我不是要算整个，我是不知道当t趋向于+∞时，limit [(-p exp(-pt) sin(wt)-w exp(-pt) cos(wt))/(p²+w²)]怎么算

追答

p>0，t->+∞，exp(-pt) = 1/exp(pt) 是无穷小量，sin(wt) 和 cos(wt) 都是 [-1,1] 内的有界函数，因此(t->+∞) lim [(-p exp(-pt) sin(wt)-w exp(-pt) cos(wt))/(p²+w²)] = 0