integral of (x^4+1)/(x^5+4x^3)

解

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

- \frac{1}{16} ln ∣ x ∣ - \frac{1}{8 x ^{2}} + \frac{17}{32} ln x^{2} + 4 + C

解答ステップ

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

以下の部分分数を得る:

\frac{x ^{4} + 1}{x ^{5} + 4 x ^{3}} : - \frac{1}{16 x} + \frac{1}{4 x ^{3}} + \frac{17 x}{16 ( x ^{2} + 4 )}

= \int - \frac{1}{16 x} + \frac{1}{4 x ^{3}} + \frac{17 x}{16 ( x ^{2} + 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 \frac{1}{16 x} d x + \int \frac{1}{4 x ^{3}} d x + \int \frac{17 x}{16 ( x ^{2} + 4 )} d x

\int \frac{1}{16 x} d x = \frac{1}{16} ln ∣ x ∣

\int \frac{1}{4 x ^{3}} d x = - \frac{1}{8 x ^{2}}

\int \frac{17 x}{16 ( x ^{2} + 4 )} d x = \frac{17}{32} ln x^{2} + 4

= - \frac{1}{16} ln ∣ x ∣ - \frac{1}{8 x ^{2}} + \frac{17}{32} ln x^{2} + 4

定数を解答に追加する

= - \frac{1}{16} ln ∣ x ∣ - \frac{1}{8 x ^{2}} + \frac{17}{32} ln x^{2} + 4 + C