 Hello and welcome to the session. In this session we will learn integration by partial fractions. We know that a rational function is the ratio of two polynomials of the form px upon qx. This qx is not equal to 0 and pxqx are the polynomials in x. Now if the degree of the polynomial px is less than the degree of the polynomial qx then the rational function is proper rational function otherwise it is called improper rational function. And the improper rational function can be reduced to proper rational function by long division process. So if the integrand is a proper rational function then it can be written as a sum of simpler rational functions by a method called partial fraction decomposition. After this the integration can be carried out easily using the already known methods. And if the integrand is improper rational function then we convert that improper rational function to a proper rational function by long division process and then we can do this by partial fraction decomposition. This table below will give us the types of simpler partial fractions that are to be associated with various kinds of rational functions like if we have the rational function of the form px plus q upon x minus a into x minus b where a is not equal to b then partial fraction will be of the form a upon x minus a plus b upon x minus b. And if the rational function is of the form px plus q upon x minus a the whole square then partial fraction would be of the form a upon x minus a plus b upon x minus a the whole square. Now next is if the rational function is of the form px square plus qx plus r upon x minus a into x minus b into x minus c then partial fraction is of the form a upon x minus a plus b upon x minus b plus c upon x minus c. Next form of rational function that we consider is px square plus qx plus r upon x minus a the whole square into x minus b then the partial fraction would be of the form a upon x minus a plus b upon x minus a the whole square plus c upon x minus b. Now the next one is rational function of the form px square plus qx plus r upon x minus a into x square plus bx plus c. Now the partial fraction here would be of the form a upon x minus a plus bx plus c upon x square plus bx plus c. In this case this x square plus bx plus c cannot be factorized further and also in the above table a b and c are real numbers to be determined suitably. Let's find out the value for this i equal to integral dx upon x plus 2 into x plus 3. The integrand that is 1 over x plus 2 into x plus 3 is a proper rational function. So by using this first form of the partial fraction we get that this integrand is equal to a over x plus 2 plus b over x plus 3. Now here the real numbers a and b are to be determined suitably. So this gives us 1 is equal to a into x plus 3 plus b into x plus 2. Now equating the coefficients of x and the constant term we get that a plus b is equal to 0 and 3a plus 2b is equal to 1. So on solving these two equations we get a is equal to 1 and b is equal to minus 1. So now the given integrand 1 over x plus 2 into x plus 3 is equal to 1 over x plus 2 minus 1 over x plus 3. So integral dx over x plus 2 into x plus 3 is equal to integral 1 over x plus 2 dx minus integral 1 over x plus 3 dx. This is equal to log modulus x plus 2 minus log modulus x plus 3 plus c which is further equal to log modulus x plus 2 over x plus 3 plus c. So this is the value of the i where the c is the constant of integration. So this completes the session. Hope you have understood how do we do the integration by partial fraction method.