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Calculus II practice

Partial Fraction Integration worksheets

Decompose proper rational functions with distinct, repeated, or irreducible quadratic factors and integrate exactly.

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Factor the denominator, include every required decomposition term, solve for coefficients, and use absolute values on linear logs.

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Each sheet asks students to expose the identity, substitution, decomposition, or limiting process that makes the integral valid. Coefficients and scales vary by seed while the selected families stay balanced.

Learning objective

What this activity builds

Build or verify a valid decomposition and integrate each rational component on domains that avoid poles.

Worked example

What terms are required for (x-r)^2?

A/(x-r) + B/(x-r)^2.

A repeated linear factor needs one term for each power through its multiplicity.

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Make the practice count

  • Check the decomposition by recombining it.
  • Mark every pole before evaluating a definite integral.
  • Separate derivative-of-quadratic and constant numerator parts.

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Math SheetNameScoreTeacherDatePartial Fraction IntegrationDecompose each proper rational function, integrate each term, and preserve absolute values on linear logarithms. Definite intervals never cross a pole.1.Evaluate the indefinite integral of (1x+0)/(x-2)^2 dx. A valid decomposition is 1/(x-2) + 2/(x-2)^2.2.Evaluate the definite integral from 0 to 1 of ((6)x-39)/((x-4)(x-7)) dx.Show the limit, decomposition, identity, or substitution that justifies each result.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDatePartial Fraction IntegrationDecompose each proper rational function, integrate each term, and preserve absolute values on linear logarithms. Definite intervals never cross a pole.3.Evaluate the indefinite integral of (7x^2-4x+40)/((x-2)(x^2+16)) dx. A valid decomposition is 3/(x-2) + (4x+4)/(x^2+16).4.Evaluate the definite integral from 0 to 1 of (4x^2-8x+9)/((x-4)(x^2+25)) dx.Show the limit, decomposition, identity, or substitution that justifies each result.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDatePartial Fraction IntegrationDecompose each proper rational function, integrate each term, and preserve absolute values on linear logarithms. Definite intervals never cross a pole.5.Evaluate the indefinite integral of (5x-16)/(x-4)^2 dx. A valid decomposition is 5/(x-4) + 4/(x-4)^2.6.Evaluate the definite integral from 0 to 1 of ((6)x-23)/((x-3)(x-8)) dx.Show the limit, decomposition, identity, or substitution that justifies each result.math-sheet.comPage 3 of 3

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