RS Aggarwal Class 12 Solutions Chapter-15 Integration Using Partial Fractions

The core principle of integration by partial fractions is to decompose a complex rational function (a fraction of two polynomials) into a sum of simpler fractions. This method works because these simpler fractions have standard, easily integrable forms. Essentially, we reverse the process of finding a common denominator. Since the integral of a sum of functions is the sum of their individual integrals, this method transforms one difficult integration problem into several simpler ones.

Vedantu provides detailed, step-by-step solutions for every question in RS Aggarwal Class 12 Chapter 15. These solutions are crafted by subject matter experts and adhere to the CBSE 2025-26 curriculum guidelines. Each solution clearly shows the process of decomposing the rational expression into partial fractions, determining the values of the constants (A, B, C, etc.), and then integrating each term separately to arrive at the final answer.

You should use integration by partial fractions specifically when the integrand is a proper rational function, which means it's a fraction where the degree of the numerator polynomial is less than the degree of the denominator polynomial. If the function doesn't fit this form, other methods are more suitable:

• Use integration by substitution when the integrand contains a function and its derivative.

• Use integration by parts when the integrand is a product of two different types of functions (e.g., algebraic and trigonometric, or logarithmic and algebraic).

According to the CBSE Class 12 syllabus, the method of integration by partial fractions is primarily applied to proper rational functions where the denominator can be factorised into:

• Non-repeated linear factors: e.g., (x-a)(x-b)

• Repeated linear factors: e.g., (x-a)²

• Non-repeated quadratic factors: e.g., (x-a)(x²+bx+c), where the quadratic factor cannot be further factorised.

A very common mistake when dealing with a repeated linear factor, such as (x-a)², in the denominator is setting up the partial fraction incorrectly. Students often write only one term, A/(x-a)², for this factor. The correct approach requires a separate fraction for each power of the repeated factor up to the highest power present. Therefore, the correct decomposition for a denominator with (x-a)² must include two terms: A/(x-a) + B/(x-a)². Forgetting the A/(x-a) term will result in an incorrect solution.

The most effective way to practise this chapter from RS Aggarwal is to first thoroughly understand the theory and solved examples. Then, approach the exercises systematically. Start with problems involving non-repeated linear factors, move to repeated linear factors, and finally tackle those with quadratic factors. Do not look at the solution immediately. Attempt to solve the problem on your own, focusing on correctly setting up the partial fractions and solving for the constants. Use the step-by-step solutions to verify your method and learn from any errors.