integral of (x^4+3x^2+5)/(x^4)

解

\int \frac{x ^{4} + 3 x ^{2} + 5}{x ^{4}} d x

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

解答ステップ

\int \frac{x ^{4} + 3 x ^{2} + 5}{x ^{4}} d x

拡張

\frac{x ^{4} + 3 x ^{2} + 5}{x ^{4}} : 1 + \frac{3}{x ^{2}} + \frac{5}{x ^{4}}

= \int 1 + \frac{3}{x ^{2}} + \frac{5}{x ^{4}} 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 1 d x + \int \frac{3}{x ^{2}} d x + \int \frac{5}{x ^{4}} d x

\int 1 d x = x

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

\int \frac{5}{x ^{4}} d x = - \frac{5}{3 x ^{3}}

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

定数を解答に追加する

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