 Hi, and welcome to the session. Let us discuss the following question. The question says, evaluate integral of 2x minus 1 by x minus 1 into x plus 2 into x minus 3 with respect to x. Now, we begin with the solution. We have to evaluate integral of 2x minus 1 by x minus 1 into x plus 2 into x minus 3 with respect to x. And here we can see that integrand is a proper rational function, because the degree of numerator is less than the degree of denominator. So by using the form of partial fractions, we can write x minus 1 divided by x minus 1 into x plus 2 into x minus 3 as a by x minus 1 plus b by x plus 2 plus c by x minus 3, where b and c are real numbers. We have to non-determine the value of ab. Now, 2x minus 1 into x plus 2 into x minus 3 equals to a by x minus 1 plus b by x plus 2 plus c by x minus 3 implies 2x minus 1 is equal to a into x plus 2 into x minus 3 plus b into x minus 1 into x minus 3 plus c into x minus 1 into x plus 2. We have got this expression by multiplying both sides of this equation by x minus 1, x plus 2, and x minus 3. We will first put as minus 2. So by substituting minus 2 in place of x, we get 2 into minus 2 minus 1 equals to a into minus 2 plus 2 into minus 2 minus 3 plus b into minus 2 minus 1 minus 2 minus 3 plus c into minus 2 minus 1 into minus 2 plus 2. Now, this implies minus 5 is equal to b into minus 3 into minus 5. This implies minus 5 is equal to 15b, and this implies b is equal to minus 153. Now, we have to go into 3 minus 1 equals to a into 3 plus 2 into 3 minus 3 plus b into 3 minus 1 into 3 minus 3 plus c into 3 minus 1 into 3 plus 2. Now, this implies 5 is equal to c into 2 into 5, and this implies c is equal to 5 by 10, and this implies c is equal to 1 by 2. By substituting 1 in place of x, we have 2 into 1 minus 1 equals to a into 1 plus 2 into 1 minus 3 plus b into 1 minus 1 into 1 minus 3 plus c into 1 minus 1 into 1 plus. Now, this implies 1 is equal to a into 3 into minus 2 plus b into 0 plus c into 0, and this implies a is equal to minus 1 by 6. Let's name this equation as equation number 1. So now, we will substitute values of a, b, and c in 1. By substituting values, we get 2x minus 1 by x minus 1 into x plus 2 into x minus 3 equals to minus 1 by 6 into 1 by x minus 1 minus 1 by 3 into 1 by x plus 2 plus 1 by 2 into 1 by x minus 3. This implies integral of 2x minus 1 by x minus 1 into x plus 2 into x minus 3 with respect to x is equal to minus 1 by 6 into integral of 1 by x minus 1 with respect to x minus 1 by 3 into integral of 1 by x plus 2 with respect to x plus 1 by 2 into integral of 1 by x minus 3 with respect to x equal to minus 1 by 6. Integral of 1 by x minus 1 with respect to x is law mod x minus 1 minus 1 by 3. Integral of 1 by x plus 2 with respect to x is law mod x plus 2 plus 1 by 2. Integral of 1 by x minus 3 with respect to x is law mod x minus 3. Plus c, where c denotes the constant of all these integrals. And the required answer is minus 1 by 6 log mod x minus 1 minus 1 by 3 log mod x plus 2 plus 1 by 2 log mod x minus 3 plus c. So this completes the session. I will take care.