sec积分

∫sec3xdx

=∫secx·sec2xdx

=∫secx·(1+tan2x)dx

=∫secxdx+∫secxtan2xdx

=ln|secx+tanx|+∫tanxdsecx

=ln|secx+tanx|+∫√(sec2x-1)dsecx

=ln|secx+tanx|+∫√(sec2x-1)dsecx

=ln|secx+tanx|+secx√(sec2x-1)/2-ln|secx+√(sec2x-1)|/2+c

首先求∫sec^3(x) dx：记I=∫sec^3(x) dx，则I

=∫sec(x)*sec^2(x) dx

=∫sec(x)*[tan(x)]' dx

=sec(x)*tan(x)-∫[sec(x)]'*tan(x) dx

=sec(x)*tan(x)-∫[sec(x)*tan(x)]*tan(x) dx

=sec(x)*tan(x)-∫sec(x)*tan^2(x) dx

=sec(x)*tan(x)-∫sec(x)*[sec^2(x)-1] dx

=sec(x)*tan(x)-∫sec^3(x) dx+∫sec(x) dx

=sec(x)*tan(x)-I+ln|sec(x)+tan(x)|+C，

所以2I=sec(x)*tan(x)+ln|sec(x)+tan(x)|+C，

I=sec(x)*tan(x)/2+ln|sec(x)+tan(x)|/2+C，C为任意常数

然后求∫sec^5(x) dx：记J=∫sec^5(x) dx，则J

=∫sec^3(x)*sec^2(x) dx

=∫sec^3(x)*[tan(x)]' dx

=sec^3(x)*tan(x)-∫[sec^3(x)]'*tan(x) dx

=sec^3(x)*tan(x)-∫3sec^2(x)*[se c(x)*tan(x)]*tan(x) dx

=sec^3(x)*tan(x)-3∫sec^3(x)*tan^2(x) dx

=sec^3(x)*tan(x)-3∫sec^3(x)*[sec^2(x)-1] dx

=sec^3(x)*tan(x)-3∫sec^5(x) dx+3∫sec^3(x) dx

=sec^3(x)*tan(x)-3J+3I，

所以4J=sec^3(x)*tan(x)+3I，

J=sec^3(x)*tan(x)/4+3I/4

=sec^3(x)*tan(x)/4+3sec(x)*tan(x)/8+3ln|sec(x)+tan(x)|/8+C，C为任意常数

%by dynamic

%see also https://www.wendangku.net/doc/3918327713.html,

%contact me matlabsky@https://www.wendangku.net/doc/3918327713.html,

%2009.2.

%

积分Matlab中提供了数值和解析解法两种

1.数值方法

matlab中所有数值积分函数的汇总比较和使用说明参见我写的教程

https://www.wendangku.net/doc/3918327713.html,/thread-359-1-3.html

>> fun=@(x)738.5*x.^2.*exp(-0.00105*x.^2);

>> quadl(fun,0,1.9)

ans =

1.6846e+003

2.符号计算

syms x

>> syms x

>> int(738.5*x^2*exp(-0.00105*x^2),x,0,1.9)

ans =

-2004500/3*exp(-7581/2000000)+52750000/63*pi^(1/2)*42^(1/2)*erf(19/20 00*42^(1/2))

>> vpa(ans)

ans =

1684.62230190651358892102680947