Exercise 3.5 — Integration by Partial Fractions

All questions in this exercise are listed below. Click on a question to view its solution.

This exercise contains 12 questions. Use the Questions tab to view and track them.

Key Concepts

This exercise focuses on integrating rational functions using the method of partial fraction decomposition.

Recognising proper vs. improper rational functions
Polynomial long division for improper fractions
Decomposing into partial fractions based on denominator type
Integrating each partial fraction term using logarithmic and power rules
Algebraic simplification and factoring of denominators
Integration of rational functions after decomposition

Important Formulas and Rules

Standard Integration Results Used

$\int \frac{1}{x} d x = ln ∣ x ∣ + C$

$\int \frac{A}{x + a} d x = A ln ∣ x + a ∣ + C$

$\int \frac{A}{( x + a ) ^{n}} d x = \frac{A ( x + a ) ^{- n + 1}}{- n + 1} + C (n \neq = 1)$

$\int \frac{1}{x ^{2} + a ^{2}} d x = \frac{1}{a} tan^{- 1} (\frac{x}{a}) + C$

Partial Fraction Decomposition Rules

Denominator Factor	Partial Fraction Form
Distinct linear: $(x + a)$	$\frac{A}{x + a}$
Repeated linear: $(x + a)^{n}$	$\frac{A _{1}}{x + a} + \frac{A _{2}}{( x + a ) ^{2}} + \dots + \frac{A _{n}}{( x + a ) ^{n}}$
Irreducible quadratic: $(a x^{2} + b x + c)$	$\frac{A x + B}{a x ^{2} + b x + c}$

Method: Step-by-Step Partial Fractions

Step 1 — Check if proper: If $de g (numerator) \geq de g (denominator)$ , perform polynomial long division first.

Step 2 — Factor the denominator completely into linear and/or irreducible quadratic factors.

Step 3 — Write the partial fraction form according to the table above.

Step 4 — Find the constants by multiplying both sides by the denominator and substituting convenient values of $x$ (or equating coefficients).

Step 5 — Integrate each term separately.

Worked Example

Evaluate $\int \frac{2 x - 1}{x ^{3} - x ^{2} - 2 x} d x$ .

Step 1: Degree of numerator (1) $<$ degree of denominator (3) — already proper.

Step 2: Factor: $x^{3} - x^{2} - 2 x = x (x^{2} - x - 2) = x (x - 2) (x + 1)$ .

Step 3: Write: $\frac{2 x - 1}{x ( x - 2 ) ( x + 1 )} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 1}$

Step 4: Multiply through by $x (x - 2) (x + 1)$ : $2 x - 1 = A (x - 2) (x + 1) + B x (x + 1) + C x (x - 2)$

$x = 0$ : $- 1 = A (- 2) (1) \Rightarrow A = \frac{1}{2}$
$x = 2$ : $3 = B (2) (3) \Rightarrow B = \frac{1}{2}$
$x = - 1$ : $- 3 = C (- 1) (- 3) \Rightarrow C = - 1$

Step 5: Integrate: $\int \frac{2 x - 1}{x ( x - 2 ) ( x + 1 )} d x = \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2∣ - ln ∣ x + 1∣ + C$

Summary

This exercise covers integration of rational functions via partial fractions. The key skill is recognising the type of denominator factor and writing the correct decomposition form before integrating.

In partial fraction decomposition, the structure of the numerator depends on the nature of the factor in the denominator.

Exercise 3.5 — Integration by Partial Fractions

All questions in this exercise are listed below. Click on a question to view its solution.

This exercise contains 12 questions. Use the Questions tab to view and track them.

Key Concepts

This exercise focuses on integrating rational functions using the method of partial fraction decomposition.

Recognising proper vs. improper rational functions
Polynomial long division for improper fractions
Decomposing into partial fractions based on denominator type
Integrating each partial fraction term using logarithmic and power rules
Algebraic simplification and factoring of denominators
Integration of rational functions after decomposition

Important Formulas and Rules

Standard Integration Results Used

$\int \frac{1}{x} d x = ln ∣ x ∣ + C$

$\int \frac{A}{x + a} d x = A ln ∣ x + a ∣ + C$

$\int \frac{A}{( x + a ) ^{n}} d x = \frac{A ( x + a ) ^{- n + 1}}{- n + 1} + C (n \neq = 1)$

$\int \frac{1}{x ^{2} + a ^{2}} d x = \frac{1}{a} tan^{- 1} (\frac{x}{a}) + C$

Partial Fraction Decomposition Rules

Denominator Factor	Partial Fraction Form
Distinct linear: $(x + a)$	$\frac{A}{x + a}$
Repeated linear: $(x + a)^{n}$	$\frac{A _{1}}{x + a} + \frac{A _{2}}{( x + a ) ^{2}} + \dots + \frac{A _{n}}{( x + a ) ^{n}}$
Irreducible quadratic: $(a x^{2} + b x + c)$	$\frac{A x + B}{a x ^{2} + b x + c}$

Method: Step-by-Step Partial Fractions

Step 1 — Check if proper: If $de g (numerator) \geq de g (denominator)$ , perform polynomial long division first.

Step 2 — Factor the denominator completely into linear and/or irreducible quadratic factors.

Step 3 — Write the partial fraction form according to the table above.

Step 4 — Find the constants by multiplying both sides by the denominator and substituting convenient values of $x$ (or equating coefficients).

Step 5 — Integrate each term separately.

Worked Example

Evaluate $\int \frac{2 x - 1}{x ^{3} - x ^{2} - 2 x} d x$ .

Step 1: Degree of numerator (1) $<$ degree of denominator (3) — already proper.

Step 2: Factor: $x^{3} - x^{2} - 2 x = x (x^{2} - x - 2) = x (x - 2) (x + 1)$ .

Step 3: Write: $\frac{2 x - 1}{x ( x - 2 ) ( x + 1 )} = \frac{A}{x} + \frac{B}{x - 2} + \frac{C}{x + 1}$

Step 4: Multiply through by $x (x - 2) (x + 1)$ : $2 x - 1 = A (x - 2) (x + 1) + B x (x + 1) + C x (x - 2)$

$x = 0$ : $- 1 = A (- 2) (1) \Rightarrow A = \frac{1}{2}$
$x = 2$ : $3 = B (2) (3) \Rightarrow B = \frac{1}{2}$
$x = - 1$ : $- 3 = C (- 1) (- 3) \Rightarrow C = - 1$

Step 5: Integrate: $\int \frac{2 x - 1}{x ( x - 2 ) ( x + 1 )} d x = \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x - 2∣ - ln ∣ x + 1∣ + C$

Summary

In partial fraction decomposition, the structure of the numerator depends on the nature of the factor in the denominator.

3.5 Exercise 3.5 — Integration by Partial Fractions

Exercise 3.5 — Integration by Partial Fractions

Key Concepts

Important Formulas and Rules

Standard Integration Results Used

Partial Fraction Decomposition Rules

Method: Step-by-Step Partial Fractions

Worked Example

Summary

3.5 Exercise 3.5 — Integration by Partial Fractions

Exercise 3.5 — Integration by Partial Fractions

Key Concepts

Important Formulas and Rules

Standard Integration Results Used

Partial Fraction Decomposition Rules

Method: Step-by-Step Partial Fractions

Worked Example

Summary