integrals - based on high math handbook for students (Справочник по высшей математике для студентов вузов / Н. Г. Тактаров)

 

Antiderivative 

F(x) otherwise function to derivative function

If y’(x) = f(x) => y’(x) is F(x) 

Indefinite integral

∫ f(x)△x = F(x)+C

*C - random constant

*△x increment

Properties

∫ k*f(x)△x = k*∫ f(x)△x

∫ (f(x) +/- g(x))△x  = ∫ f(x)△x +/- ∫ g(x)△x  

Integral sum

∑[i=1,n] f(s) △x

*△x = x(i) - x(i-1)

*∑ - sum between

Definite integral

S = lim( △x->0 ) ∑[i=1,n] f(s) △x = ∫ [a,b] f(x)△x

*as we see geometrical understanding of definete integral is area under the plot

Properties

∫ [a,a] f(x)△x = 0

∫ [a,a] f(x)△x = - ∫ [a,a] f(x)△x

∫ [a,c] f(x)△x = ∫ [a,b] f(x)△x + ∫ [b,c] f(x)△x

∫ [a,b] (f(x) +/- g(x))△x = ∫ [a,b] f(x)△x +/- ∫ [a,b] g(x)△x

∫ [a,c] k*f(x)△x = k*∫ [a,c] f(x)△x

If f(x)<>=g(x) then ∫ [a,b] f(x)△x <>= ∫ [a,b] g(x)△x