Reduction Formula

Derive the reduction formula below:

    \[\displaystyle \int x^{m}\cdot(\ln {x})^n \,\mathrm{d}x = \frac{x^{m+1}}{m+1} \cdot (\ln {x})^n - \frac{n}{m+1} \int x^{m}\cdot(\ln {x})^{n-1} \,\mathrm{d}x\]

Source:

Edwards, C.H., & Penney, D.E. (1986). Calculus and Analytic Geometry (2nd ed.). Englewood Cliffs, NJ: Prentice-Hall. (Problems 9.4, Question 54, p. 469)

Author: ascklee

Dr. Lee teaches at the Wee Kim Wee School of Communication and Information at the Nanyang Technological University in Singapore. He founded The Mathematics Digital Library in 2013.

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