integral of sqrt(x)-2/(x^5)+7/x

解

\int x - \frac{2}{x ^{5}} + \frac{7}{x} d x

\frac{2}{3} x^{\frac{3}{2}} + \frac{1}{2 x ^{4}} + 7 ln ∣ x ∣ + C

解答ステップ

\int x - \frac{2}{x ^{5}} + \frac{7}{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 x d x - \int \frac{2}{x ^{5}} d x + \int \frac{7}{x} d x

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

\int \frac{2}{x ^{5}} d x = - \frac{1}{2 x ^{4}}

\int \frac{7}{x} d x = 7 ln ∣ x ∣

= \frac{2}{3} x^{\frac{3}{2}} - (- \frac{1}{2 x ^{4}}) + 7 ln ∣ x ∣

簡素化

= \frac{2}{3} x^{\frac{3}{2}} + \frac{1}{2 x ^{4}} + 7 ln ∣ x ∣

定数を解答に追加する

= \frac{2}{3} x^{\frac{3}{2}} + \frac{1}{2 x ^{4}} + 7 ln ∣ x ∣ + C