integral of (3(-5+ln(4x))^3)/x

解

\int \frac{3 ( - 5 + ln ( 4 x ) ) ^{3}}{x} d x

3 (- 125 ln ∣ x ∣ + \frac{75}{2} ln^{2} (4 x) - 5 ln^{3} (4 x) + \frac{ln ^{4} ( 4 x )}{4}) + C

解答ステップ

\int \frac{3 ( - 5 + ln ( 4 x ) ) ^{3}}{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 \frac{( - 5 + ln ( 4 x ) ) ^{3}}{x} d x

拡張

\frac{( - 5 + ln ( 4 x ) ) ^{3}}{x} : - \frac{125}{x} + \frac{75 ln ( 4 x )}{x} - \frac{15 ln ^{2} ( 4 x )}{x} + \frac{ln ^{3} ( 4 x )}{x}

= 3 \cdot \int - \frac{125}{x} + \frac{75 ln ( 4 x )}{x} - \frac{15 ln ^{2} ( 4 x )}{x} + \frac{ln ^{3} ( 4 x )}{x} d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 (- \int \frac{125}{x} d x + \int \frac{75 ln ( 4 x )}{x} d x - \int \frac{15 ln ^{2} ( 4 x )}{x} d x + \int \frac{ln ^{3} ( 4 x )}{x} d x)

\int \frac{125}{x} d x = 125 ln ∣ x ∣

\int \frac{75 ln ( 4 x )}{x} d x = \frac{75}{2} ln^{2} (4 x)

\int \frac{15 ln ^{2} ( 4 x )}{x} d x = 5 ln^{3} (4 x)

\int \frac{ln ^{3} ( 4 x )}{x} d x = \frac{ln ^{4} ( 4 x )}{4}

= 3 (- 125 ln ∣ x ∣ + \frac{75}{2} ln^{2} (4 x) - 5 ln^{3} (4 x) + \frac{ln ^{4} ( 4 x )}{4})

定数を解答に追加する

= 3 (- 125 ln ∣ x ∣ + \frac{75}{2} ln^{2} (4 x) - 5 ln^{3} (4 x) + \frac{ln ^{4} ( 4 x )}{4}) + C