Using a Table of Integrals

Use a Table of Integrals to find \(\int \dfrac{dx}{\sqrt{\left(4x-x^{2}\right) ^{3}}}.\)

Solution Look through the headings in the Table of Integrals until you locate Integrals Containing \(\sqrt{2ax-x^{2}}.\) Continue in the subsection until you find a form that closely resembles the integrand given. You should find Integral 82: \[ \int \dfrac{dx}{(2ax-x^{2}) ^{3/2}}=\dfrac{x-a}{a^{2}\sqrt{2ax-x^{2}}}+C \] This is the integral we seek with \(a = 2.\) So, \[ \int \dfrac{dx}{\sqrt{( 4x-x^{2}) ^{3}}}=\dfrac{x-2}{4\sqrt{4x-x^{2}}}+C \]