用数学软件Maple计算定积分（例题与习题）..doc

用数学软件Maple计算定积分（例题与习题） 2012/6/15 例 题 1. a:=n-n*sum(1/(n^2+k^2),k=1..n); Limit(a(n),n=infinity)=limit(a(n),n=infinity); 2. limit(integrate(arctan(x)^2/x,x=n..n+1),n=infinity); 3 limit(integrate(x^n/(1+x),x=0..1),n=infinity); f[n_]:=Integrate[x^n/(1+x),{x,0,1}]; Limit[f[n],n?Infinity] 5. f:=x-sqrt(1-sin(x)): a:=0:b:=Pi: Integrate(f(x),x=a..b)=integrate(f(x),x=a..b); 6. f:=n-integrate(tan(x)^n,x=0..Pi/4); f(n)+f(n-2); f:=n-integrate(tan(x)^n,x=0..Pi/4); f(4)+f(2); f(5)+f(3); f(10)+f(8); f(9)+f(7); 8 F:=(x,y)-integrate(exp(t^2),t=0..x)+integrate(cos(t^2),t=0..y)+sin(x+y); Fx:=diff(F(x,y),x); Fy:=diff(F(x,y),y); DyDx:=-Fx/Fy; 9 x:=t-2*t^2; y:=t-int(exp(u)/u,u=1..1+2*ln(t)); t0:=2: Dx:=D(x)(t0): Dy:=D(y)(t0): Dxx:=(D@@2)(x)(t0): Dyy:=(D@@2)(y)(t0): Yijie=Dy/Dx: simplify(%); Erjie=(Dx*Dyy-Dxx*Dy)/Dx^3: simplify(%); x[t_]:=2*t^2; y[t_]:=Integrate[Exp[u]/u,{u,1,1+2*Log[t]}]; Y[t_]=y[t]/x[t]; Er[t_]=Y[t]/x[t]; Y[2]; Simplify[%] Er[2]; Simplify[%] 11 f:=x-(sin(x)-cos(x))/(1+sin(x)*cos(x)): a:=0:b:=Pi/2: Integrate(f(x),x=a..b)=integrate(f(x),x=a..b); 16. f:=x-integrate(exp(-t^2),t=0..x^2); plot(f(x),x=-2..3,thickness=2); f:=x-integrate(exp(-t^2),t=0..x^2); diff(f(x),x); solve(diff(f(x),x),x); diff(f(x),x$2); solve(diff(f(x),x$2),x); 17（1） f:=x-sin(x)/x: F:=Integrate(f(t),t=1..x^2): Diff(F,x)=diff(F,x); with(plots): f:=x-int(sin(t)/t,t=1..x^2); quxian:=plot(f(x),x=-5.5..5.5,thickness=3): display(quxian,scaling=unconstrained); （2） f:=x-integrate(sin(t)/t,t=1..x^2): a:=0:b:=1: Integrate(x*f(x),x=a..b)=integrate(x*f(x),x=a..b); 20 f:=x-int(sin(t^2),t=0..sin(x))/x^3; Limit(f(x),x=0)=limit(f(x),x=0); 21 with(plots): f:=x-max(x,x^2,x^3); plot(f(x),x=-2..2,thickness=3); with(plots): f:=x-max(x,x^2,x^3); F:=x-int(f(t),t=0..x); plot(F(x),x=-2..2,thickness=3,scaling=constrained); 22 f:=x-x*exp(x)/(1+exp(x))^2: Integrate(f(x),x=0..infinity)=integrate(f(x),x=0..infinity); 23 a:=7: f:=x-1/(1+x^2)/(1+x^a): Integrate(f(x),x=0