integral of ((5+ln(x))^5)/x

解

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

3125 ln ∣ x ∣ + \frac{3125}{2} ln^{2} (x) + \frac{1250}{3} ln^{3} (x) + \frac{125}{2} ln^{4} (x) + 5 ln^{5} (x) + \frac{ln ^{6} ( x )}{6} + C

解答ステップ

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

拡張

\frac{( 5 + ln ( x ) ) ^{5}}{x} : \frac{3125}{x} + \frac{3125 ln ( x )}{x} + \frac{1250 ln ^{2} ( x )}{x} + \frac{250 ln ^{3} ( x )}{x} + \frac{25 ln ^{4} ( x )}{x} + \frac{ln ^{5} ( x )}{x}

= \int \frac{3125}{x} + \frac{3125 ln ( x )}{x} + \frac{1250 ln ^{2} ( x )}{x} + \frac{250 ln ^{3} ( x )}{x} + \frac{25 ln ^{4} ( x )}{x} + \frac{ln ^{5} ( 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

= \int \frac{3125}{x} d x + \int \frac{3125 ln ( x )}{x} d x + \int \frac{1250 ln ^{2} ( x )}{x} d x + \int \frac{250 ln ^{3} ( x )}{x} d x + \int \frac{25 ln ^{4} ( x )}{x} d x + \int \frac{ln ^{5} ( x )}{x} d x

\int \frac{3125}{x} d x = 3125 ln ∣ x ∣

\int \frac{3125 ln ( x )}{x} d x = \frac{3125}{2} ln^{2} (x)

\int \frac{1250 ln ^{2} ( x )}{x} d x = \frac{1250}{3} ln^{3} (x)

\int \frac{250 ln ^{3} ( x )}{x} d x = \frac{125}{2} ln^{4} (x)

\int \frac{25 ln ^{4} ( x )}{x} d x = 5 ln^{5} (x)

\int \frac{ln ^{5} ( x )}{x} d x = \frac{ln ^{6} ( x )}{6}

= 3125 ln ∣ x ∣ + \frac{3125}{2} ln^{2} (x) + \frac{1250}{3} ln^{3} (x) + \frac{125}{2} ln^{4} (x) + 5 ln^{5} (x) + \frac{ln ^{6} ( x )}{6}

定数を解答に追加する

= 3125 ln ∣ x ∣ + \frac{3125}{2} ln^{2} (x) + \frac{1250}{3} ln^{3} (x) + \frac{125}{2} ln^{4} (x) + 5 ln^{5} (x) + \frac{ln ^{6} ( x )}{6} + C