PART 26：例題-綜合除法假分式(求不定積分)

\(\int {\frac{{3{x^4} - 6x + 3}}{{{{(x + 1)}^3}}}dx} \)

SOL：

(1)以綜合除法將分子的 \(x\) 改成 \((x + 1)\) 表示(要注意補0)

\(3{x^4} - 6x + 3\)

\( = 3{(x + 1)^4}\) \( - 12{(x + 1)^3}\) \( + 18{(x + 1)^2}\) \( - 18(x + 1)\) \( + 12\)

(2)拆開原式積分成為5項

\(\int {\frac{{3{x^4} - 6x + 3}}{{{{(x + 1)}^3}}}dx} \)

\( = \int {\frac{{3{{(x + 1)}^4} - 12{{(x + 1)}^3} + 18{{(x + 1)}^2} - 18(x + 1) + 12}}{{{{(x + 1)}^3}}}dx} \)

\( = \int {\frac{{3{{(x + 1)}^4} - 12{{(x + 1)}^3} + 18{{(x + 1)}^2} - 18(x + 1) + 12}}{{{{(x + 1)}^3}}}dx} \)

\( = \int {3(x + 1)dx}  - \int {12dx}  + \int {\frac{{18}}{{x + 1}}dx}  - \int {\frac{{18}}{{{{(x + 1)}^2}}}dx}  + \int {\frac{{12}}{{{{(x + 1)}^3}}}dx} \)

\( = \frac{3}{2}{(x + 1)^2} - 12x + 18\ln (x + 1) + \frac{{18}}{{x + 1}} - \frac{6}{{{{(x + 1)}^2}}} + C\)