integral of (4/x+2x^3-2/(x^3))

解

\int (\frac{4}{x} + 2 x^{3} - \frac{2}{x ^{3}}) d x

4 ln ∣ x ∣ + \frac{x ^{4}}{2} + \frac{1}{x ^{2}} + C

解答ステップ

\int \frac{4}{x} + 2 x^{3} - \frac{2}{x ^{3}} 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{4}{x} d x + \int 2 x^{3} d x - \int \frac{2}{x ^{3}} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int 2 x^{3} d x = \frac{x ^{4}}{2}

\int \frac{2}{x ^{3}} d x = - \frac{1}{x ^{2}}

= 4 ln ∣ x ∣ + \frac{x ^{4}}{2} - (- \frac{1}{x ^{2}})

簡素化

= 4 ln ∣ x ∣ + \frac{x ^{4}}{2} + \frac{1}{x ^{2}}

定数を解答に追加する

= 4 ln ∣ x ∣ + \frac{x ^{4}}{2} + \frac{1}{x ^{2}} + C