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

解

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

x^{3} - x^{2} + x + \frac{1}{x} + C

解答ステップ

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

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

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

\int 1 d x = x

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

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

簡素化

= x^{3} - x^{2} + x + \frac{1}{x}

定数を解答に追加する

= x^{3} - x^{2} + x + \frac{1}{x} + C