PART 23：例題-配方(求不定積分)

\(\int {\frac{{3x + 5}}{{{x^2} + 2x + 2}}dx} \)

SOL：(分母不能分解)

(1)分母配方
\(\int {\frac{{3x + 5}}{{{x^2} + 2x + 2}}dx} \) \( = \int {\frac{{3x + 5}}{{({x^2} + 2x + 1) + 1}}dx} \) \( = \int {\frac{{3x + 5}}{{{{(x + 1)}^2} + 1}}dx} \)

(2)變數變換
令 \(u = x + 1\) ，\(du = dx\)
原式 \( = \int {\frac{{3u + 2}}{{{u^2} + 1}}du} \) \( = \int {\frac{{3u}}{{{u^2} + 1}}du}  + \int {\frac{2}{{{u^2} + 1}}du} \)  \( = \frac{3}{2}\ln ({u^2} + 1) + 2{\tan ^{ - 1}}u + C\)

(3)還原為 \(x\)
原式 \( = \frac{3}{2}\ln ({u^2} + 1) + 2{\tan ^{ - 1}}u + C\) \( = \frac{3}{2}\ln ({x^2} + 2x + 2) + 2{\tan ^{ - 1}}(x + 1) + C\)