integral of x^2(1/(x^3)+1/(sqrt(x)))

解

\int x^{2} (\frac{1}{x ^{3}} + \frac{1}{x}) d x

ln ∣ x ∣ + \frac{2}{5} x^{\frac{5}{2}} + C

解答ステップ

\int x^{2} (\frac{1}{x ^{3}} + \frac{1}{x}) d x

拡張

x^{2} (\frac{1}{x ^{3}} + \frac{1}{x}) : \frac{1}{x} + x^{\frac{3}{2}}

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

\int \frac{1}{x} d x = ln ∣ x ∣

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

= ln ∣ x ∣ + \frac{2}{5} x^{\frac{5}{2}}

定数を解答に追加する

= ln ∣ x ∣ + \frac{2}{5} x^{\frac{5}{2}} + C