1)
ʃdz/(5z+1)³
u = 5z+1
du/dz = (5z+1)' = 5
dz = du/5
ʃdz/(5z+1)³ = ʃdu/5u³ =(1/5)ʃdu/u³ =-1/5*2u² =-1/10u² = -1/10(5z+1)² + c
2)
ʃdx/√(3x-1)³
u = 3x-1
du/dx = (3x-1)' = 3
dx = du/3
ʃdx/√(3x-1)³ = (1/3)ʃdu/u^(3/2) = (1/3)(-2/√u) =-2/3√u = -2/3√(3x-1) + c
3)
ʃx³dx/(5x^4+3)⁵
du/dx = (5x^4+3)' = 20x³
dx = du/20x³
ʃx³dx/(5x^4+3)⁵ = ʃx³du/(20x³)*u⁵ = (1/20)ʃdu/u⁵ =-(1/20)(1/4u^4) =
=-1/80u^4 = -1/80(5x^4+3)^4 + c
4)
ʃcos(x)dx/√(1-sin(x))
u = 1-sin(x)
du/dx = (1-sin(x))' = - cos(x)
dx = - du/cos(x)
ʃcos(x)dx/√(1-sin(x)) = -ʃcos(x)du/cos(x)√u = -ʃdu/√u = -2√u = -2√(1-sin(x)) + c
5)
ʃx³dx/³√(5x^4 +2²)²
u = 5x^4 +4
du/dx = (5x^4 +4)' = 20x³
dx = du/20x³
ʃx³dx/³√(5x^4 +2²)² = ʃx³du/20x³u^(2/3) = (1/20)ʃdu/u^(2/3) = (1/20)*3*³√u=
= (3*³√u)/20 = (3*³√(5x^4 +4))/20+ c