integral of (3-3x^2)ln(x)

解

\int (3 - 3 x^{2}) ln (x) d x

3 x ln (x) - x^{3} ln (x) - 3 x + \frac{x ^{3}}{3} + C

解答ステップ

\int (3 - 3 x^{2}) ln (x) d x

(3 - 3 x^{2}) ln (x) = 3 (1 - x^{2}) ln (x)

= \int 3 (1 - x^{2}) ln (x) d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int (1 - x^{2}) ln (x) d x

部分積分を適用する

= 3 (ln (x) (x - \frac{x ^{3}}{3}) - \int \frac{1}{3} (3 - x^{2}) d x)

\int \frac{1}{3} (3 - x^{2}) d x = x - \frac{x ^{3}}{9}

= 3 (ln (x) (x - \frac{x ^{3}}{3}) - (x - \frac{x ^{3}}{9}))

簡素化

3 (ln (x) (x - \frac{x ^{3}}{3}) - (x - \frac{x ^{3}}{9})) : 3 x ln (x) - x^{3} ln (x) - 3 x + \frac{x ^{3}}{3}

= 3 x ln (x) - x^{3} ln (x) - 3 x + \frac{x ^{3}}{3}

定数を解答に追加する

= 3 x ln (x) - x^{3} ln (x) - 3 x + \frac{x ^{3}}{3} + C