Various Method of Integrations

Integration in mathematics this comes under the category of calculus. It is a basic concept of calculus. Integration means calculation of area which forms between X-axis and Y-axis of any function. Two basic theorems are used for integral calculus. First theorem is the integral of the product of a constant and a function is equal to product of the constant and the integral of function. Second theorem is the integral of a sum of or difference of a finite number of functions is equal to sum or difference of integral. Trigonometric identities are very useful to solving integration problems. There are various methods of integration by which we can reduce the given integral to one of the known standard integrals. There are four principal methods of integration.

What is integration by parts?
Integration by part means integral of the product of two functions. Suppose we have two functions of X are U and V. then we use differential calculus and integrate both sides. Finally we get the formula for integration by part. The choice of which function will be U and which function will be dV is very important in solving integration by part. The ILATE method helps to decide this. ILATE stands for
I: inverse trigonometric function
L: logarithmic function
A: algebraic function
T: trigonometric function
E: exponential function

Whichever of the two functions comes first in ILATE, get designated as U and other function designated as dV.

Integration by parts example
First example, suppose two functions of X are algebraic and trigonometric function. By using ILATE method first integrate algebraic function till it becomes zero. Now integrate trigonometric function. We have to write the solution such as first term of algebraic function multiply with second term of trigonometric function and similarly next term.

Second example, suppose two functions of X are trigonometric and exponential function. We have to repeat same procedure but here we integrate trigonometric function first then exponential function, because in ILATE, trigonometric function comes first then exponential function.
Similarly we can take any two functions product for integration. These above are two very simple integration by parts examples.

Now we discuss about integration by parts tabular method, tabular integration by parts is vary valuable method to finding integrals. Tabular method can also be used for derivations of some important theorems in analysis. Tabular integration means successive integration by parts. In this method there are two columns, in first column function and its derivatives and in second column function and its anti-derivatives.