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人気のある微分積分 >

integral of (32x^2)/(x^4-34x^2+225)

解

\int \frac{32 x ^{2}}{x ^{4} - 34 x ^{2} + 225} d x

解

3 ln ∣ x + 3 ∣ - 3 ln ∣ x - 3 ∣ - 5 ln ∣ x + 5 ∣ + 5 ln ∣ x - 5 ∣ + C

解答ステップ

\int \frac{32 x ^{2}}{x ^{4} - 34 x ^{2} + 225} d x

定数を除く:

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

= 32 \cdot \int \frac{x ^{2}}{x ^{4} - 34 x ^{2} + 225} d x

以下の部分分数を得る:

\frac{x ^{2}}{x ^{4} - 34 x ^{2} + 225} : \frac{3}{32 ( x + 3 )} - \frac{3}{32 ( x - 3 )} - \frac{5}{32 ( x + 5 )} + \frac{5}{32 ( x - 5 )}

= 32 \cdot \int \frac{3}{32 ( x + 3 )} - \frac{3}{32 ( x - 3 )} - \frac{5}{32 ( x + 5 )} + \frac{5}{32 ( x - 5 )} 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

= 32 (\int \frac{3}{32 ( x + 3 )} d x - \int \frac{3}{32 ( x - 3 )} d x - \int \frac{5}{32 ( x + 5 )} d x + \int \frac{5}{32 ( x - 5 )} d x)

\int \frac{3}{32 ( x + 3 )} d x = \frac{3}{32} ln ∣ x + 3 ∣

\int \frac{3}{32 ( x - 3 )} d x = \frac{3}{32} ln ∣ x - 3 ∣

\int \frac{5}{32 ( x + 5 )} d x = \frac{5}{32} ln ∣ x + 5 ∣

\int \frac{5}{32 ( x - 5 )} d x = \frac{5}{32} ln ∣ x - 5 ∣

= 32 (\frac{3}{32} ln ∣ x + 3 ∣ - \frac{3}{32} ln ∣ x - 3 ∣ - \frac{5}{32} ln ∣ x + 5 ∣ + \frac{5}{32} ln ∣ x - 5 ∣)

簡素化

32 (\frac{3}{32} ln ∣ x + 3 ∣ - \frac{3}{32} ln ∣ x - 3 ∣ - \frac{5}{32} ln ∣ x + 5 ∣ + \frac{5}{32} ln ∣ x - 5 ∣) : 3 ln ∣ x + 3 ∣ - 3 ln ∣ x - 3 ∣ - 5 ln ∣ x + 5 ∣ + 5 ln ∣ x - 5 ∣

= 3 ln ∣ x + 3 ∣ - 3 ln ∣ x - 3 ∣ - 5 ln ∣ x + 5 ∣ + 5 ln ∣ x - 5 ∣

定数を解答に追加する

= 3 ln ∣ x + 3 ∣ - 3 ln ∣ x - 3 ∣ - 5 ln ∣ x + 5 ∣ + 5 ln ∣ x - 5 ∣ + C

グラフ

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