StudyBrixFree worksheets, study guides and courses

Mathematics · University

Integration by parts with a logarithm

Here n > −1. Treat x = 0 as an improper endpoint and use its limiting boundary term.

Make a worksheet

Method and conditions

Here n > −1. Treat x = 0 as an improper endpoint and use its limiting boundary term.

∫₀¹ xⁿ ln(x) dx = −1/(n + 1)²

Worked examples

Evaluate the improper integral from 0 to 1 of 9 x^1 ln(x) dx. Round the final result to 4 decimal places if needed.

Use u = ln x and dv = 9x^1 dx. The boundary term vanishes; the remaining integral is −9/(2)².

-2.25

Evaluate the improper integral from 0 to 1 of 8 x^12 ln(x) dx. Round the final result to 4 decimal places if needed.

Use u = ln x and dv = 8x^12 dx. The boundary term vanishes; the remaining integral is −8/(13)².

-0.0473

Try it

  1. Evaluate the improper integral from 0 to 1 of 13 x^16 ln(x) dx. Round the final result to 4 decimal places if needed.

  2. Evaluate the improper integral from 0 to 1 of 9 x^6 ln(x) dx. Round the final result to 4 decimal places if needed.

Check your work

Answers and marking notes
  1. -0.045

    Use u = ln x and dv = 13x^16 dx. The boundary term vanishes; the remaining integral is −13/(17)².

  2. -0.1837

    Use u = ln x and dv = 9x^6 dx. The boundary term vanishes; the remaining integral is −9/(7)².

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.

AI-assisted preview; independent educational review required. Level labels describe prerequisites, not accreditation or official curriculum alignment.