
- •Vector space. The linear dependence of vectors. Decomposition of the vector on the basis.
- •Balance sheet ratio
- •The linear model of a diversified economy
- •Functions of one variable. The concept of the function. Limit function. The concept of continuity of a function. Points of discontinuity of functions.
- •The concept of a function of several variables
- •Gradient
- •Indefinite integral. Basic properties. Table of basic indefinite integrals. The concept of primitive function
- •The indefinite integral
- •Basic properties of the indefinite integral
- •Integration by Parts
- •Definite integral. Basic properties. The basic formula of integral calculus. Basic rules of integration. Applying of integrals in economy.
- •Classes of integrable functions
- •The basic properties of the definite integral
- •The basic formula of integral calculus
- •The basic rules of integration Replacing a variable in the definite integral
- •Integration by parts in a definite integral
The basic formula of integral calculus
This formula is called the basic formula of integral calculus, or the Newton-Leibniz formula.
The Newton-Leibniz formula gives ample opportunities calculate definite integrals. It is necessary to compute the indefinite integral and then find the difference in the values of the primitive functions. Consider the examples of the calculation of definite integrals.
Example 1.
Example 2.
The basic rules of integration Replacing a variable in the definite integral
THEOREM. Let: 1) a continuous function on [a, b]; 2) the function φ (t) is differentiable on [α, β], where φ '(t) is continuous on [α, β] and the set of values of the function φ (t) is the interval [a, b], 3) φ (α) = a, φ (β) = b. Then the formula
Formula (7.12) called the formula replacing of the variable or substitution of the variable in a definite integral.
Calculate definite integrals by substitution.
Example 3.
DECISION.
Substitute t
=
1 + х2.
Then dt
= 2х
dx, t
= 1
for х
= 0
and t
=
2
for x = 1. Since the function
х
=
is continuous on [1, 2], and the new integrand is also continuous,
and hence into force for it Theorem there is a primitive on this
segment. Get
Integration by parts in a definite integral
Theorem. Let u (x) and v (x) have continuous derivatives on [a, b]; Then the formula
Equality (7.13) is a formula for integration by parts in a definite integral. Consider some examples of calculating definite integrals by integration by parts.
Example 4.
DECISION. Put here и = х, v = e-x, тогда dv = -e-xdx и