Integration by parts

Methods of integration
I- Integration by parts
Sometimes we can recognize the differential to be integrated as a product of a function which is easily differentiated and a differential which is easily integrated.
For example, if the problem is to find the integral

We have two way to do this:
1- Tabular integration by parts
Which is used for
1) the products of polynomials and sine function,
2) the products of polynomials and cosine function,
3) the products of polynomials and exponential function, .
In any of these three cases we choose the polynomial as and the product of sine function and (cosine or exponential function and dx – respectively) as .
Example1: Evaluate



Methods of integration
I- Integration by parts
Sometimes we can recognize the differential to be integrated as a product of a function which is easily differentiated and a differential which is easily integrated.
For example, if the problem is to find the integral

We have two way to do this:
1- Tabular integration by parts
Which is used for
1) the products of polynomials and sine function,
2) the products of polynomials and cosine function,
3) the products of polynomials and exponential function, .
In any of these three cases we choose the polynomial as and the product of sine function and (cosine or exponential function and dx – respectively) as .
Example1: Evaluate







2- Integration by using the formula

Which is used for inverse trigonometric functions and logarithm functions.
Example2: Evaluate




























2- Integration by using the formula

Which is used for inverse trigonometric functions and logarithm functions.
Example2: Evaluate