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

解

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

\frac{2}{5} x^{\frac{5}{2}} + \frac{2 x ^{3}}{3} - \frac{x ^{2}}{2} - \frac{2}{x} + C

解答ステップ

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

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

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

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

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

= \frac{2}{5} x^{\frac{5}{2}} + \frac{2 x ^{3}}{3} - \frac{x ^{2}}{2} - \frac{2}{x}

定数を解答に追加する

= \frac{2}{5} x^{\frac{5}{2}} + \frac{2 x ^{3}}{3} - \frac{x ^{2}}{2} - \frac{2}{x} + C