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人気のある微分積分 >

integral of (2x^3+11x+8)/(x^3+4x^2+4x)

解

\int \frac{2 x ^{3} + 11 x + 8}{x ^{3} + 4 x ^{2} + 4 x} d x

解

2 x - 10 ln ∣ x + 2 ∣ - \frac{15}{x + 2} + 2 ln ∣ x ∣ + 4 + C

解答ステップ

\int \frac{2 x ^{3} + 11 x + 8}{x ^{3} + 4 x ^{2} + 4 x} d x

因数

\frac{2 x ^{3} + 11 x + 8}{x ^{3} + 4 x ^{2} + 4 x}

= \int \frac{2 x ^{3} + 11 x + 8}{x ( x + 2 ) ^{2}} d x

拡張

\frac{2 x ^{3} + 11 x + 8}{x ( x + 2 ) ^{2}} : \frac{2 x ^{3}}{x ( x + 2 ) ^{2}} + \frac{11 x}{x ( x + 2 ) ^{2}} + \frac{8}{x ( x + 2 ) ^{2}}

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{2 x ^{3}}{x ( x + 2 ) ^{2}} d x + \int \frac{11 x}{x ( x + 2 ) ^{2}} d x + \int \frac{8}{x ( x + 2 ) ^{2}} d x

\int \frac{2 x ^{3}}{x ( x + 2 ) ^{2}} d x = 2 (x + 2 - 4 ln ∣ x + 2 ∣ - \frac{4}{x + 2})

\int \frac{11 x}{x ( x + 2 ) ^{2}} d x = - \frac{11}{x + 2}

\int \frac{8}{x ( x + 2 ) ^{2}} d x = 8 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{2 ( x + 2 )})

= 2 (x + 2 - 4 ln ∣ x + 2 ∣ - \frac{4}{x + 2}) - \frac{11}{x + 2} + 8 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{2 ( x + 2 )})

簡素化

2 (x + 2 - 4 ln ∣ x + 2 ∣ - \frac{4}{x + 2}) - \frac{11}{x + 2} + 8 (\frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 2 ∣ + \frac{1}{2 ( x + 2 )}) : 2 x - 10 ln ∣ x + 2 ∣ - \frac{15}{x + 2} + 2 ln ∣ x ∣ + 4

= 2 x - 10 ln ∣ x + 2 ∣ - \frac{15}{x + 2} + 2 ln ∣ x ∣ + 4

定数を解答に追加する

= 2 x - 10 ln ∣ x + 2 ∣ - \frac{15}{x + 2} + 2 ln ∣ x ∣ + 4 + C

グラフ

人気の例

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w \to 0 lim (\frac{a - w ^{2} + a ^{2}}{b - w ^{2} + b ^{2}})

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