Unit 6: Partial Fractions of Indefinite Integrals

Partial fractions decomposition is a crucial technique in calculus used to simplify complex rational functions by breaking them down into simpler fractions.  For example,  [latex]\frac{3x+1}{x^2-4x+3}=\frac{2}{x-1}+\frac{1}{x+3}[/latex].

This method is particularly useful for solving integration problems involving rational functions, as it allows us to express the integrand as a sum of simpler fractions, making the integration process more manageable. For example, [latex]\int \frac{3x+1}{x^2-4x+3}dx=\int \frac{2}{x-1}dx +\int \frac{1}{x+3}dx[/latex].

The simpler fractions on the right side of the above equations are typically decomposed into linear factors, repeated linear factors, and irreducible quadratic factors, with their numerators having lower degrees than the denominator. These decompositions can be categorized into four cases, ranging from easier to more challenging.

Practice Questions:

Evaluate the following indefinite integrals before referring to the video for comprehensive solutions.

• Case 1: the denominator of the rational function is the product of multiple linear functions

[latex]\frac{}{(ax+b)(cx+d)...}=\frac{A}{ax+b}+\frac{B}{cx+d} + ...[/latex]

Q1: [latex]\int \frac{5x-3}{x^2-2x-3}dx[/latex]

Q2: [latex]\int \frac{x^2+4x+1}{(x+1)(x-1)(x+3)}dx[/latex]

• Case 2: the denominator of the rational function is the product of multiple linear functions and the power of a linear function

[latex]\frac{}{(ax+b)(cx+d)(ex+f)^n}=\frac{A}{ax+b}+\frac{B}{cx+d} + \frac{C_1}{ex+f} +\frac{C_2}{(ex+f)^2} ... +\frac{C_n}{(ex+f)^n}[/latex]

Q3: [latex]\int \frac{3x^2+7x+8}{x^3+4x^2+4x}dx[/latex]

• Case 3: the denominator of the rational function is the product of multiple linear functions and a quadratic function

[latex]\frac{}{(ax+b)(cx+d)(ex^2+f)}=\frac{A}{ax+b}+\frac{B}{cx+d} + \frac{Cx+D}{ex^2+f}[/latex]

                 [latex]\frac{}{(ax+b)(cx+d)^n(ex^2+f)}=\frac{A}{ax+b}+ + \frac{B_1}{cx+d} +\frac{B_2}{(cx+d)^2} ... +\frac{B_n}{(cx+d)^n} + \frac{Cx+D}{ex^2+f}[/latex]

Q4: [latex]\int \frac{-2t+4}{(t^2+1)(t-1)^2}dt[/latex]

• Case 4: the denominator of the rational function is the product of multiple linear functions and multiple quadratic functions

[latex]\frac{}{(ax+b)(cx+d)(ex^2+f)^n}=\frac{A}{ax+b}+\frac{B}{cx+d} + \frac{C_1}{ex+f} +\frac{C_2}{(ex+f)^2} ... +\frac{C_n}{(ex+f)^n}[/latex]

[latex]\frac{}{(ax+b)(cx+d)^m(ex^2+f)^n}=\frac{A}{ax+b}+ \frac{B_1}{cx+d} +... +\frac{B_m}{(cx+d)^m} + \frac{C_1}{ex+f}+... +\frac{C_n}{(ex+f)^n}[/latex]

Q5: [latex]\int \frac{1}{y(y^2+1)^2}dy[/latex]