第8章 8.14

（証明）

$I_n = \int (\log x)^n dx = x(\log x)^n - \int x \cdot n(\log x)^{n-1} \cdot \frac{1}{x} dx = x(\log x)^n - n\int (\log x)^{n-1} dx$

$= x(\log x)^n - nI_{n-1} \qquad (n \geq 1)$

$n = 0$ のときは

$I_0 = \int (\log x)^0 dx = \int dx = x + C$

(1)

$\int (\log x)^2 dx$ は 8.12(2) より

$= x(\log x)^2 - 2x \cdot \log x + 2x + C \quad \text{なので}$

$\int (\log x)^3 dx = x(\log x)^3 - 3\int (\log x)^2 dx = x(\log x)^3 - 3\{x(\log x)^2 - 2x \cdot \log x + 2x + C\}$

$= x(\log x)^3 - 3x(\log x)^2 + 6x \cdot \log x - 6x + C$

(2)

$\int (\log x)^4 dx = x(\log x)^4 - 4\int (\log x)^3 dx$

$= x(\log x)^4 - 4\{x(\log x)^3 - 3x(\log x)^2 + 6x \cdot \log x - 6x + C\}$

$= x(\log x)^4 - 4x(\log x)^3 + 12x(\log x)^2 - 24x \cdot \log x + 24x + C$