数学实验 第三章

第三章练习题

1.

（1）syms t;

f=exp(5*t);

int(f,t)

运行结果：

ans =1/5*exp(5*t)

（2）syms x;

f=x/(x^4+2*x^2+5);

int(f,x)

运行结果：

ans =1/4*atan(1/2*x^2+1/2)

（3）syms x;

f=1/(x*log(x)*log(log(x))); int(f,x)

运行结果：

ans =log(log(log(x)))

（4）syms x;

f=sqrt(1+cos(x));

int(f,x)

运行结果：ans

=-2*cos(1/2*x)*(-1+cos(1/2*x)^2)/sin(1 /2*x)*2^(1/2)/(cos(1/2*x)^2)^(1/2)

（5）syms x a;

f=x^2/(sqrt(a^2-x^2));

int(f,x)

运行结果：

ans

=-1/2*x*(a^2-x^2)^(1/2)+1/2*a^2*atan (x/(a^2-x^2)^(1/2))

（6）syms x;

f=1/(1+sqrt(2*x));

int(f,x)

运行结果：

ans

=-1/2*log(2*x-1)+2^(1/2)*x^(1/2)-atanh (2^(1/2)*x^(1/2))

（7）syms x;

f=cos(x)*sin(x)/(1+cos(x)^2);

int(f,x)

运行结果：

ans =-1/2*log(1+cos(x)^2)

（8）syms x;

f=1/(x*sqrt(x^2-1));

int(f,x)

运行结果：

ans =-atan(1/(x^2-1)^(1/2)) （9）syms x;

f=sqrt(x^2-9)/x;

int(f,x)

运行结果：

ans

=(x^2-9)^(1/2)+3*atan(3/(x^2-9)^(1/2))

2.

（1）syms x ;

f=x*exp(-x^2/2);

int(f,x,0,1)

运行结果：

ans =-exp(-1/2)+1

（2）syms x ;

f=(abs(x)+x)*exp(-x^2); int(f,x,-2,2)

运行结果：

ans = -exp(-4)+1

（3）syms x ;

f=abs(x-1);

int(f,x,0,2)

运行结果：

ans = 1 （4）syms x ;

f=1/(exp(x)+1);

int(f,x,0,1)

运行结果：

ans =1-log(1+exp(1))+log(2)

（5）syms x ;

f=sqrt(1+cos(2*x));

int(f,x,0,pi)

运行结果：

ans = 2*2^(1/2)

（6）syms x ;

f=1/(x^2*sqrt(1+x^2));

int(f,x,1,sqrt(3))

运行结果：

ans =-2/3*3^(1/2)+2^(1/2)

3.

（1）syms x t;

f=acos(t^2);

diff(int(f,t,1,x),x)

运行结果为;

Ans= = (acos(x^2)*(1-x^4)^(1/2)-2*x^4*acos(x^ 2)/(1-x^4)^(1/2)+2/(1-x^2)^(1/2)*(1+x^ 2)^(1/2)*EllipticF(x,i)*x-2*(1-x^2)^(1/2) /(1+x^2)^(1/2)*EllipticF(x,i)*x-2/(1-x^2) ^(1/2)*(1+x^2)^(1/2)*EllipticE(x,i)*x+2* (1-x^2)^(1/2)/(1+x^2)^(1/2)*EllipticE(x,i )*x-4*EllipticK(i)/(1-x^4)^(1/2)*x^3+4*E llipticE(i)/(1-x^4)^(1/2)*x^3)/(1-x^4)^(1 /2)+2*(x*acos(x^2)*(1-x^4)^(1/2)-2*(1-x^2)^(1/2)*(1+x^2)^(1/2)*EllipticF(x,i)+ 2*(1-x^2)^(1/2)*(1+x^2)^(1/2)*EllipticE( x,i)+2*EllipticK(i)*(1-x^4)^(1/2)-2*Ellipti cE(i)*(1-x^4)^(1/2))/(1-x^4)^(3/2)*x^3

（2）syms x t;

f=sqrt(1+t^4);

diff(int(f,t,0,log(x)),x)

运行结果为;

ans=(1/12-1/12*i)*(1/x*2^(1/2)+5*log(x)^4*2^(1/2)/x+(2+3*i)/x*2^(1/2)+5*i*log(x)^4*2^ (1/2)/x-4*i/(1-i*log(x)^2)^(1/2)*(1+i*log(x)^2 )^(1/2)*EllipticF((1/2+1/2*i)*log(x)*2^(1/2),i) *log(x)/x+4*i*(1-i*log(x)^2)^(1/2)/(1+i*log(x) ^2)^(1/2)*EllipticF((1/2+1/2*i)*log(x)*2^(1/2) ,i)*log(x)/x)*2^(1/2)/(1+log(x)^4)^(1/2)+(-1/6 +1/6*i)*(log(x)*2^(1/2)+log(x)^5*2^(1/2)+i*l og(x)*2^(1/2)+i*log(x)^5*2^(1/2)+4*(1-i*log( x)^2)^(1/2)*(1+i*log(x)^2)^(1/2)*EllipticF((1/ 2+1/2*i)*log(x)*2^(1/2),i))*2^(1/2)/(1+log(x) ^4)^(3/2)*log(x)^3/x

（3）syms x t;

f=sin(t^2+1);

diff(int(f,t,-x^2,1),x)

运行结果为;

2*cos(1)*sin(x^4)*x+2*sin(1)*cos(x^4)* x

（4）syms x t;

f=sqrt(1+log(t));

diff(int(f,t,exp(x),x),x)

运行结果为;

(1+log(x))^(1/2)-exp(x)*(1+log(exp(x)))^( 1/2)