求上面黄框内的积分

>> syms x a t s kesi
>> f=kesi*s/(1+kesi*kesi)^2
 
f =
 
(kesi*s)/(kesi^2 + 1)^2
 
>> mider=int(f,kesi,x-a*(t-s),x+a*(t-s))
 
mider =
 
s/(2*((x + a*(s - t))^2 + 1)) - s/(2*((x - a*(s - t))^2 + 1))
 
>> finall=int(mider,s,0,t)
 
finall =
 
int(s/(2*((x + a*(s - t))^2 + 1)) - s/(2*((x - a*(s - t))^2 + 1)), s, 0, t)
 
>> int(s/(2*((x + a*(s - t))^2 + 1)) - s/(2*((x - a*(s - t))^2 + 1)), s, 0, t)
 
ans =
 
int(s/(2*(x + a*(s - t))^2 + 2) - s/(2*(x - a*(s - t))^2 + 2), s, 0, t)
 
>> simplify(ans)
 
ans =
 
int(s/(2*(x + a*(s - t))^2 + 2) - s/(2*(x - a*(s - t))^2 + 2), s, 0, t)
 
>> int(s/(1+(x+a*(t-s))^2),s,0,t)
 
ans =
 
int(s/((x - a*(s - t))^2 + 1), s, 0, t)
 
>> syms y,eqn1=x+a*(t-s)==y
 
eqn1 =
 
x - a*(s - t) == y
 
>> solve(eqn1,x)
 
ans =
 
y + a*(s - t)
 
>> solve(eqn1,s)
 
ans =
 
(x - y + a*t)/a
 
>> int_1f=ans/(1+y*y)
 
int_1f =
 
(x - y + a*t)/(a*(y^2 + 1))
 
>> symplify(int_1f)
函数或变量 'symplify' 无法识别。
 
是不是想输入:
>> simplify(int_1f)
 
ans =
 
(x - y + a*t)/(a*(y^2 + 1))
 
>> int_1=int(ans,x+a*t,x)
 
int_1 =
 
-(t*(2*x - 2*y + 3*a*t))/(2*(y^2 + 1))
 
>> subs(int_1,y,x - a*(s - t) )
 
ans =
 
-(t*(3*a*t + 2*a*(s - t)))/(2*((x - a*(s - t))^2 + 1))
 
>> int_1=int(simplify(int_1f),y,x+a*t,x)
 
int_1 =
 
- t*(atan(x + a*t) - atan(x)) - (log(x^2 + 1)/2 - log((x + a*t)^2 + 1)/2 + x*(atan(x + a*t) - atan(x)))/a
 
>> simplify(int_1)
 
ans =
 
- t*(atan(x + a*t) - atan(x)) - (log(x^2 + 1)/2 - log((x + a*t)^2 + 1)/2 + x*(atan(x + a*t) - atan(x)))/a
 
>> syms y2,eqn2=x-a*(t-s)==y2
 
eqn2 =
 
x + a*(s - t) == y2
 
>> solve(eqn2,s)
 
ans =
 
(y2 - x + a*t)/a
 
>> int_2f=ans/(1+y*y)
 
int_2f =
 
(y2 - x + a*t)/(a*(y^2 + 1))
 
>> int_2=int(symplify(int_2f),y2,x-a*t,x)
函数或变量 'symplify' 无法识别。
 
是不是想输入:
>> int_2=int(simplify(int_2f),y2,x-a*t,x)
 
int_2 =
 
(a*t^2)/(2*(y^2 + 1))
 
>> fff=latex(symplify(-(int_1-int_2)/(4*a)))
函数或变量 'symplify' 无法识别。
 
是不是想输入:
>> fff=latex(simplify(-(int_1-int_2)/(4*a)))

fff =

    '\frac{t\,\left(\mathrm{atan}\left(x+a\,t\right)-\mathrm{atan}\left(x\right)\right)+\frac{\frac{\ln\left(x^2+1\right)}{2}-\frac{\ln\left({\left(x+a\,t\right)}^2+1\right)}{2}+x\,\left(\mathrm{atan}\left(x+a\,t\right)-\mathrm{atan}\left(x\right)\right)}{a}+\frac{a\,t^2}{2\,\left(y^2+1\right)}}{4\,a}'

>> 

注意到

>> finall=int(mider,s,0,t)
 
finall =
 
int(s/(2*((x + a*(s - t))^2 + 1)) - s/(2*((x - a*(s - t))^2 + 1)), s, 0, t)
 
>> int(s/(2*((x + a*(s - t))^2 + 1)) - s/(2*((x - a*(s - t))^2 + 1)), s, 0, t)

推测是分母太过于复杂，这时需要将被积函数换元。定义新变量y=x±a*(t-s)，分别处理。（所有的步骤都贴在开始的代码中）