詳解：為了遷就分母，以綜合除法將分子之 \(x\) 改為 \((x - 2)\) 表達

\(\int {\frac{{4{x^3} - 2x + 7}}{{{{(x - 2)}^4}}}\;dx}  = \int {\frac{{4{{(x - 2)}^3} + 24{{(x - 2)}^2} + 46(x - 2) + 35}}{{{{(x - 2)}^4}}}} dx\)

\( = \int {\frac{{4{{(x - 2)}^3}}}{{{{(x - 2)}^4}}}} dx + \int {\frac{{24{{(x - 2)}^2}}}{{{{(x - 2)}^4}}}} dx + \int {\frac{{46(x - 2)}}{{{{(x - 2)}^4}}}} dx + \int {\frac{{35}}{{{{(x - 2)}^4}}}} dx\)

上下有相同因式部份可對消

\( = \int {\frac{4}{{(x - 2)}}} dx + \int {\frac{{24}}{{{{(x - 2)}^2}}}} dx + \int {\frac{{46}}{{{{(x - 2)}^3}}}} dx + \int {\frac{{35}}{{{{(x - 2)}^4}}}} dx\)

\( = \int {\frac{4}{{(x - 2)}}} dx + \int {24{{(x - 2)}^{ - 2}}dx + } \int {46{{(x - 2)}^{ - 3}}} dx + \int {35{{(x - 2)}^{ - 4}}} dx\)

以基本積分公式求出積分

\( = 4\ln (x - 2) + \frac{{24}}{{ - 1}}{(x - 2)^{ - 1}} + \frac{{46}}{{ - 2}}{(x - 2)^{ - 2}} + \frac{{35}}{{ - 3}}{(x - 2)^{ - 3}} + C\)

\( = 4\ln (x - 2) - \frac{{24}}{{x - 2}} - \frac{{23}}{{{{(x - 2)}^2}}} - \frac{{35}}{{3{{(x - 2)}^3}}} + C\)