Mathematics · University
Integration by parts with a logarithm · formula reference
Here n > −1. Treat x = 0 as an improper endpoint and use its limiting boundary term.
Formula
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 example
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
Prerequisites and checking
Algebra; calculus or linear algebra as appropriate to the topic. Keep units and rounding consistent. State the model conditions before applying the formula.
Further reading and review references
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