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人気のある微分積分 >

integral of (22x^2)/(x^4-61x^2+900)

解

\int \frac{22 x ^{2}}{x ^{4} - 61 x ^{2} + 900} d x

解

5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 6 ln ∣ x + 6 ∣ + 6 ln ∣ x - 6 ∣ + C

解答ステップ

\int \frac{22 x ^{2}}{x ^{4} - 61 x ^{2} + 900} d x

定数を除く:

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

= 22 \cdot \int \frac{x ^{2}}{x ^{4} - 61 x ^{2} + 900} d x

以下の部分分数を得る:

\frac{x ^{2}}{x ^{4} - 61 x ^{2} + 900} : \frac{5}{22 ( x + 5 )} - \frac{5}{22 ( x - 5 )} - \frac{3}{11 ( x + 6 )} + \frac{3}{11 ( x - 6 )}

= 22 \cdot \int \frac{5}{22 ( x + 5 )} - \frac{5}{22 ( x - 5 )} - \frac{3}{11 ( x + 6 )} + \frac{3}{11 ( 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

= 22 (\int \frac{5}{22 ( x + 5 )} d x - \int \frac{5}{22 ( x - 5 )} d x - \int \frac{3}{11 ( x + 6 )} d x + \int \frac{3}{11 ( x - 6 )} d x)

\int \frac{5}{22 ( x + 5 )} d x = \frac{5}{22} ln ∣ x + 5 ∣

\int \frac{5}{22 ( x - 5 )} d x = \frac{5}{22} ln ∣ x - 5 ∣

\int \frac{3}{11 ( x + 6 )} d x = \frac{3}{11} ln ∣ x + 6 ∣

\int \frac{3}{11 ( x - 6 )} d x = \frac{3}{11} ln ∣ x - 6 ∣

= 22 (\frac{5}{22} ln ∣ x + 5 ∣ - \frac{5}{22} ln ∣ x - 5 ∣ - \frac{3}{11} ln ∣ x + 6 ∣ + \frac{3}{11} ln ∣ x - 6 ∣)

簡素化

22 (\frac{5}{22} ln ∣ x + 5 ∣ - \frac{5}{22} ln ∣ x - 5 ∣ - \frac{3}{11} ln ∣ x + 6 ∣ + \frac{3}{11} ln ∣ x - 6 ∣) : 5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 6 ln ∣ x + 6 ∣ + 6 ln ∣ x - 6 ∣

= 5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 6 ln ∣ x + 6 ∣ + 6 ln ∣ x - 6 ∣

定数を解答に追加する

= 5 ln ∣ x + 5 ∣ - 5 ln ∣ x - 5 ∣ - 6 ln ∣ x + 6 ∣ + 6 ln ∣ x - 6 ∣ + C

グラフ

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