Табличный интеграл
[b] ∫ 2^(u)du=2^(u)/ln2 + C
[/b]
u=2x-3
du=2dx
d(2x-3)=2dx
dx=(1/2)d(2x-3)
∫ ^(1)_(0)2^(2x-3)dx=(1/2) ∫ ^(1)_(0)2^(2x-3)d(2x-3)=
= (1/2)*(2^(2x-3)/ln2)|^(1)_(0)=(1/2)(2^(-1)/ln2)-(1/2)2^(-3)/ln2=
= [b]3/(16ln2)[/b]
2)
Табличный интеграл
[b] ∫ u^(2)du=u^(3)/3 + C
[/b]
u=(6-x)
du= - dx
d(6-x)=-dx
dx=-d(6-x)
∫ ^(1)_(-1)(6-x)^(2)dx=- ∫ ^(1)_(-1)(6-x)^(2)d(6-x)=
= - ((6-x)^3/3)|^(1)_(-1)=-(5^3/3)+(7^3/3)=(7^3-5^3)/3=...
3)
Табличный интеграл
[b] ∫ sinu du=-cosu + C
[/b]
u=(x/3)
du= (1/3) dx
d(x/3)=(1/3)dx
dx=3d(x/3)
∫ ^(3π)_(π/2) sin(x/3)dx=3 ∫ ^(3π)_(π/2) sin(x/3)d(x/3)=
=-3cos(x/3) | ^(3π)_(π/2)= -3* cos(π)+3cos(π/6)=-3*(-1)+3*(1/2)=4,5