Partial Fractions

Partial Fractions Method for Indefinite Integrals:

  • Principle: Used for decomposing a complex rational function into simpler fractions.
  • Applicability: Typically applied when integrating rational functions (ratio of polynomials).
  • Steps:

Steps for Partial Fractions Decomposition:

  1. Proper Fraction:

    • Ensure that the degree of the numerator is less than the degree of the denominator.
  2. Factorization:

    • Factorize the denominator into irreducible factors.
  3. Partial Fractions Decomposition:

    • Express the given rational function as a sum of partial fractions.
    • If the factors are linear and distinct: P(x)Q(x)=Axr1+Bxr2++Nxrn
    • If there are repeated factors: P(x)Q(x)=A1xr1+A2(xr1)2++Ak(xr1)k
  4. Determining Coefficients:

    • Find the unknown coefficients by various methods like equating coefficients, cover-up method, or simultaneous equations.
  5. Integration:

    • Integrate each partial fraction term individually.

Example:

Let's solve the indefinite integral ∫ (4x + 7) / (x2+ 3x + 2) dx using the partial fractions method:

Steps:

  1. Proper Fraction Check:

    • The degree of the numerator (1) is less than the degree of the denominator (2). It's a proper fraction.
  2. Factorization:

    • Factorize the denominator: x2+3x+2=(x+1)(x+2).
  3. Partial Fractions Decomposition:

    • Express (4x+7)/(x2+3x+2) as:

    4x+7x2+3x+2=Ax+1+Bx+2

  4. Determining Coefficients:

    • Multiply both sides by the denominator to solve for A and B.
    • 4x+7=A(x+2)+B(x+1)
    • Substitute values of x to solve for A and B. (e.g., x = -1 and x = -2)

    Solving for A and B gives: A = 3 and B = 1.

  5. Integration:

    • Rewrite the integral in terms of the partial fractions:

    4x+7x2+3x+2dx=3x+1+1x+2dx

    • Integrate each term individually:

    =3lnx+1+lnx+2+C

Tips for Partial Fractions:

  • Factorization is Key: Properly factorize the denominator before decomposing into partial fractions.
  • Systematic Approach: Write down the partial fraction form and solve for unknown coefficients methodically.
  • Check your Work: Ensure the decomposition is correct by recombining the partial fractions.