How to solve definite integrals

The solution of the definite integral always comes down to bringing his original expression to the tabular view, which is easy to calculate. The main problem is search of ways of the cast.

Common solutions

Repeat a textbook of mathematical analysis or higher mathematics that is definite integrals. As is known, the solution of the definite integral is a function whose derivative will give the integrand. This function is called primitive. On this principle the table of basic integrals.
Determine by referring to the integrand which a table of integrals is suitable in this case. It is not always possible to determine immediately. Often, the table view becomes visible only after several transformations to simplify the integrand.

Method of change of variables

If the expanded function is trigonometric function, the argument which some polynomial, then try to use the method of change of variables. In order to do this, replace the polynomial at the argument of the integrand, for some new variable. The ratio between the old and new variable define the new limits of integration. Differentiation of this expression, locate the new differential in the integral. Thus, you will get a new form of the previous integral is close to or even corresponding to any table.

A solution of integrals of the second kind

If the integral is the integral of the second kind, which means the vector form of the integrand, you will need to use the rules of moving from data to scalar integrals. One of these rules is the ratio of Gauss. The law lets go of the rotor flux of some vector function to the triple integral of the divergence of this vector field.

Substitution of limits of integration

After finding the integral you need to substitute the limits of integration. First substitute the upper limit value in the expression for the integral. You will get some number. Then subtract from the resulting number and another number obtained by substituting the lower limit in the integral. If one of the limits of integration is infinity, then by substituting it in the integral must go to the limit and find what seeks expression.
If the integral is two-dimensional or three-dimensional, you have to portray geometrically the limits of integration to understand how to calculate the integral. Indeed, in the case of, say, the three-dimensional integrals the limits of integration may be the whole plane bounding the volume integrated.