Technique explanation
This lesson applies Paravartya Yojayet (Transpose and Adjust) to partial fraction decomposition and establishes the boundary for its use in calculus.
Paravartya allows finding the numerators of partial fractions by substitution. For a fraction f(x)/((x-a)(x-b)), the numerator for (x-a) is found by substituting x=a into the rest of the expression. This is a direct application of "transpose and adjust."
This method is used to decompose rational functions before integration. However, the Vedic method only optimizes the algebraic setup; the actual integration follows standard calculus rules. This establishes a clear boundary between Vedic algebra and calculus execution.
Standard fallback: The method of undetermined coefficients, which involves setting up a system of linear equations by equating the coefficients of like powers of x to solve for the unknown numerators.
Worked examples
Example 1: Decompose (3x+5)/((x+1)(x+2))
- Step 1: To find A for (x+1), set x=-1 in (3x+5)/(x+2): A = (3(-1)+5)/(-1+2) = 2/1 = 2.
- Step 2: To find B for (x+2), set x=-2 in (3x+5)/(x+1): B = (3(-2)+5)/(-2+1) = -1/-1 = 1.
- Result: 2/(x+1) + 1/(x+2).
- Check: Recombine the partial fractions: 2/(x+1) + 1/(x+2) = [2(x+2) + 1(x+1)] / [(x+1)(x+2)] = (2x + 4 + x + 1) / [(x+1)(x+2)] = (3x + 5) / [(x+1)(x+2)].
Example 2: Set up ∫ (x²+1)/((x-1)(x-2)(x-3)) dx
- Step 1: Decompose the integrand. x=1 → A = (1+1)/((1-2)(1-3)) = 2/2 = 1.
- Step 2: x=2 → B = (4+1)/((2-1)(2-3)) = 5/-1 = -5.
- Step 3: x=3 → C = (9+1)/((3-1)(3-2)) = 10/2 = 5.
- Setup: ∫ (1/(x-1) - 5/(x-2) + 5/(x-3)) dx. The integration is now a standard term-by-term step.
- Check: Verify the numerator expansion: (x-2)(x-3) - 5(x-1)(x-3) + 5(x-1)(x-2) = (x²-5x+6) - 5(x²-4x+3) + 5(x²-3x+2) = x² - 5x + 6 - 5x² + 20x - 15 + 5x² - 15x + 10 = x² + 1.
Academy Course