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Wize University Calculus 2 Textbook > Integration Techniques

Partial Fractions

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Partial Fractions Decomposition

Identifying Clues

  • When we have a rational function P(x)Q(x)\frac{P\left(x\right)}{Q\left(x\right)}

Strategy

Wize Concept

Case 1: The denominator only has linear factors


Case 2: The denominator has quadratic factors



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Steps


Practice Question

Compute ∫3x2−1x4+x3  ⁣dx{\displaystyle\int}\frac{3x^2-1}{x^4+x^3}\de{x}.

Practice Question

Evaluate ∫x4+x3+6x2+2x+4x(x2+2)2  ⁣dx.{\displaystyle\int}\frac{x^4+x^3+6x^2+2x+4}{x(x^2+2)^2}\de{x}.

Practice Question

Find the integral ∫2x3−3x+2x3−x2−2x  ⁣dx{\displaystyle\int}\frac{2x^3-3x+2}{x^3-x^2-2x}\de{x}
Extra Practice
Integration by Partial Fractions
Evaluate the integral
∫x+4x2−5x+6dx\int_{ }^{ }\frac{x+4}{x^2-5x+6}dx
Use upper case CC to denote any constants.
Example: Partial Fractions
Example: Partial Fractions (w/ Repeated Factors)
Evaluate ∫x4+x3+6x2+2x+4x(x2+2)2  ⁣dx.{\displaystyle\int}\frac{x^4+x^3+6x^2+2x+4}{x(x^2+2)^2}\de{x}.
Example: Partial Fractions
Example: Partial Fractions (w/ Repeated Factors)
Evaluate ∫x4+x3+6x2+2x+4x(x2+2)2  ⁣dx.{\displaystyle\int}\frac{x^4+x^3+6x^2+2x+4}{x(x^2+2)^2}\de{x}.
Integration by Partial Fractions
Evaluate the integral
∫x+4x2−5x+6dx\int_{ }^{ }\frac{x+4}{x^2-5x+6}dx
Use upper case CC to denote any constants.
Integration by Partial Fractions
Evaluate the integral
∫x+4x2−5x+6dx\int_{ }^{ }\frac{x+4}{x^2-5x+6}dx
Use upper case CC to denote any constants.
Integration by Partial Fractions
Evaluate the integral
∫x+4x2−5x+6dx\int_{ }^{ }\frac{x+4}{x^2-5x+6}dx
Use upper case CC to denote any constants.
Example: Partial Fractions
Example: Partial Fractions (w/ Repeated Factors)
Evaluate ∫x4+x3+6x2+2x+4x(x2+2)2  ⁣dx.{\displaystyle\int}\frac{x^4+x^3+6x^2+2x+4}{x(x^2+2)^2}\de{x}.
Example: Partial Fractions
Example: Partial Fractions (w/ Repeated Factors)
Evaluate ∫x4+x3+6x2+2x+4x(x2+2)2  ⁣dx.{\displaystyle\int}\frac{x^4+x^3+6x^2+2x+4}{x(x^2+2)^2}\de{x}.
Integration by Partial Fractions
Evaluate the integral
∫x+4x2−5x+6dx\int_{ }^{ }\frac{x+4}{x^2-5x+6}dx
Use upper case CC to denote any constants.
Example: Partial Fractions
Example: Partial Fractions (w/ Repeated Factors)
Evaluate ∫x4+x3+6x2+2x+4x(x2+2)2  ⁣dx.{\displaystyle\int}\frac{x^4+x^3+6x^2+2x+4}{x(x^2+2)^2}\de{x}.