Mathematics in Economics – lecture 9

Indefinite integral

Integration is a reverse procedure to differentiation.

Notation:

Legend:            …. Integration sign – indefinite integral;      f(x) …. Integrated function;

F(x) … antiderivative of f(x); …C …. Integration constant

Indefinite integral is a linear operator:


We compute integrals with the use of formulas above, and with the use of the table of elementary
integrals:

Indefinite integral – elementary integrals


Indefinite integral - examples

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Indefinite integral – integration methods

For more complicated integration we use suitable integration methods:

•         Substitutions

•          Method per partes

All these methods will be demonstrated on examples.

I)                  Integration by a substitution

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HOMEWORK

A]

B]

C]

D]

E]                       F]

II)              Integration by parts (per partes method)

Per partes method (integration by parts) is used for integration of a product of two functions.

Let u(x) and v(x) be two functions. Then, we obtain:

The last formula is “per partes“ formula.

Note: a choice of u and v´ is important. An incorrect choice leads to a growing difficulty of a
problem.


2)







HOMEWORK

A]

B]

C]