Задача 35144 ...
Условие
математика ВУЗ
525
Решение
★
Замена переменной
[b]x^(3/5)=t[/b]
d(x^(3/5)=dt
dt=(x^(3/5))`*dx
dt=(3/5)*x^(-2/5)dx
dt=(3/5)*(dx/x^(2/5))
[b]dx/x^(2/5)=(5/3)dt[/b]
∫ ctg(x^(3/5))dx/x^(2/5)=(5/3) ∫ ctgtdt=(5/3) ∫ costdt/sint=
=(5/3) ∫ d(sint)/cost=(5/3)ln|cost|+C= [b](5/3)ln|cos(x^(3/5))|+C[/b]
2.
Замена переменной
[b]sin3x=t[/b]
d(sin3x)=dt
(sin3x)`dx=dt
cos3x*(3x)`dx=dt
3cos3xdx=dt
[b]cos3xdx=dt/3[/b]
∫ сos3xdx/(25+sin^23x)= ∫ (dt/3)/(25+t^2)=(1/3)*(1/5)arctg(t/5)+C=
= [b](1/15)arctg((sin3x)/5)+C[/b]
3.
Замена переменной
[b]1/x^2=t[/b]
d(1/x^2)=dt
(-2/x^3)dx=dt
[b](2/x^3)dx=-dt[/b]
∫ (2/x^3)*tg(1/x^2)dx=- ∫ tgtdt= - ∫ sintdt/cost= ∫ d(cost)/sint=
=ln}sint|+C = [b] ln |tg(1/x^2)|+C[/b]