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

解

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

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

解答ステップ

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

拡張

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

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

\int 1 d x = x

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

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

定数を解答に追加する

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