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人気のある微分積分 >

integral of (3x^39x^2-15)/(x^24x^4)

解

\int \frac{3 x ^{3} 9 x ^{2} - 15}{x ^{2} 4 x ^{4}} d x

解

\frac{1}{4} (27 ln ∣ x ∣ + \frac{3}{x ^{5}}) + C

解答ステップ

\int \frac{3 x ^{3} \cdot 9 x ^{2} - 15}{x ^{2} \cdot 4 x ^{4}} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int \frac{3 x ^{3} \cdot 9 x ^{2} - 15}{x ^{2} x ^{4}} d x

簡素化

\frac{3 x ^{3} \cdot 9 x ^{2} - 15}{x ^{2} x ^{4}} : \frac{27 x ^{5} - 15}{x ^{6}}

= \frac{1}{4} \cdot \int \frac{27 x ^{5} - 15}{x ^{6}} d x

拡張

\frac{27 x ^{5} - 15}{x ^{6}} : \frac{27}{x} - \frac{15}{x ^{6}}

= \frac{1}{4} \cdot \int \frac{27}{x} - \frac{15}{x ^{6}} 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

= \frac{1}{4} (\int \frac{27}{x} d x - \int \frac{15}{x ^{6}} d x)

\int \frac{27}{x} d x = 27 ln ∣ x ∣

\int \frac{15}{x ^{6}} d x = - \frac{3}{x ^{5}}

= \frac{1}{4} (27 ln ∣ x ∣ - (- \frac{3}{x ^{5}}))

簡素化

= \frac{1}{4} (27 ln ∣ x ∣ + \frac{3}{x ^{5}})

定数を解答に追加する

= \frac{1}{4} (27 ln ∣ x ∣ + \frac{3}{x ^{5}}) + C

グラフ

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