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

解

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

x^{2} - ln ∣ x ∣ + ln (x) + cot (ln (x)) + C

解答ステップ

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

拡張

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

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

\int 2 x d x = x^{2}

\int \frac{csc ^{2} ( ln ( x ) )}{x} d x = ln ∣ x ∣ - ln (x) - cot (ln (x))

= x^{2} - (ln ∣ x ∣ - ln (x) - cot (ln (x)))

簡素化

= x^{2} - ln ∣ x ∣ + ln (x) + cot (ln (x))

定数を解答に追加する

= x^{2} - ln ∣ x ∣ + ln (x) + cot (ln (x)) + C