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人気のある微分積分 >

integral of (x+3)/(2x^3-8x)

解

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

解

\frac{1}{2} (- \frac{3}{4} ln ∣ x ∣ + \frac{1}{8} ln ∣ x + 2 ∣ + \frac{5}{8} ln ∣ x - 2 ∣) + C

解答ステップ

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

因数

\frac{x + 3}{2 x ^{3} - 8 x}

= \int \frac{x + 3}{2 x ( x + 2 ) ( x - 2 )} d x

定数を除く:

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

= \frac{1}{2} \cdot \int \frac{x + 3}{x ( x + 2 ) ( x - 2 )} d x

以下の部分分数を得る:

\frac{x + 3}{x ( x + 2 ) ( x - 2 )} : - \frac{3}{4 x} + \frac{1}{8 ( x + 2 )} + \frac{5}{8 ( x - 2 )}

= \frac{1}{2} \cdot \int - \frac{3}{4 x} + \frac{1}{8 ( x + 2 )} + \frac{5}{8 ( 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

= \frac{1}{2} (- \int \frac{3}{4 x} d x + \int \frac{1}{8 ( x + 2 )} d x + \int \frac{5}{8 ( x - 2 )} d x)

\int \frac{3}{4 x} d x = \frac{3}{4} ln ∣ x ∣

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

\int \frac{5}{8 ( x - 2 )} d x = \frac{5}{8} ln ∣ x - 2 ∣

= \frac{1}{2} (- \frac{3}{4} ln ∣ x ∣ + \frac{1}{8} ln ∣ x + 2 ∣ + \frac{5}{8} ln ∣ x - 2 ∣)

定数を解答に追加する

= \frac{1}{2} (- \frac{3}{4} ln ∣ x ∣ + \frac{1}{8} ln ∣ x + 2 ∣ + \frac{5}{8} ln ∣ x - 2 ∣) + C

グラフ

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