integral of ((1+ln(x))^2)/(sqrt(x))

解

\int \frac{( 1 + ln ( x ) ) ^{2}}{x} d x

10 x - 4 x ln (x) + 2 x ln^{2} (x) + C

解答ステップ

\int \frac{( 1 + ln ( x ) ) ^{2}}{x} d x

拡張

\frac{( 1 + ln ( x ) ) ^{2}}{x} : \frac{1}{x} + \frac{2 ln ( x )}{x} + \frac{ln ^{2} ( x )}{x}

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

\int \frac{1}{x} d x = 2 x

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

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

= 2 x + 2 (2 x ln (x) - 4 x) + 2 x ln^{2} (x) - 4 (2 x ln (x) - 4 x)

簡素化

2 x + 2 (2 x ln (x) - 4 x) + 2 x ln^{2} (x) - 4 (2 x ln (x) - 4 x) : 10 x - 4 x ln (x) + 2 x ln^{2} (x)

= 10 x - 4 x ln (x) + 2 x ln^{2} (x)

定数を解答に追加する

= 10 x - 4 x ln (x) + 2 x ln^{2} (x) + C