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Calculus II practice

Improper Integrals worksheets

Classify and evaluate infinite-bound and singular integrals using one-sided limits, including interior splits.

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Write every required limit. Split at an interior singularity and require both one-sided pieces to converge.

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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

Replace every improper feature by the correct limit and distinguish genuine convergence from divergence without principal values.

Worked example

Can symmetric infinities cancel across an interior pole?

No, not in the ordinary improper integral.

The left and right one-sided integrals must converge separately; a principal value is a different concept.

Teaching notes

Make the practice count

  • Locate all infinite bounds and denominator zeros first.
  • Use p-integral tests before evaluating.
  • Keep convergence classification visible next to the value.

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Improper cases
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Math SheetNameScoreTeacherDateImproper IntegralsRewrite each integral as the required one-sided limit, split at every interior singularity, and classify convergence before evaluating. Do not use principal values.1.Classify and, if convergent, evaluate integral from 0 to 3 of 3/sqrt(x) dx.2.Classify and, if convergent, evaluate integral from -4 to 4 of 3/|x| dx. Split at x = 0; do not use a principal value.Show the limit, decomposition, identity, or substitution that justifies each result.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDateImproper IntegralsRewrite each integral as the required one-sided limit, split at every interior singularity, and classify convergence before evaluating. Do not use principal values.3.Classify and, if convergent, evaluate integral from 0 to 3 of 3/x dx.4.Classify and, if convergent, evaluate integral from 4 to infinity of 2/x^2 dx.Show the limit, decomposition, identity, or substitution that justifies each result.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDateImproper IntegralsRewrite each integral as the required one-sided limit, split at every interior singularity, and classify convergence before evaluating. Do not use principal values.5.Classify and, if convergent, evaluate integral from -3 to 3 of 3/sqrt(|x|) dx. Split at x = 0; do not use a principal value.6.Classify and, if convergent, evaluate integral from 3 to infinity of 3/x^1 dx.Show the limit, decomposition, identity, or substitution that justifies each result.math-sheet.comPage 3 of 3

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