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人気のある微分積分 >

integral of (66x^2)/(x^4-65x^2+784)

解

\int \frac{66 x ^{2}}{x ^{4} - 65 x ^{2} + 784} d x

解

4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣ + C

解答ステップ

\int \frac{66 x ^{2}}{x ^{4} - 65 x ^{2} + 784} d x

定数を除く:

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

= 66 \cdot \int \frac{x ^{2}}{x ^{4} - 65 x ^{2} + 784} d x

以下の部分分数を得る:

\frac{x ^{2}}{x ^{4} - 65 x ^{2} + 784} : \frac{2}{33 ( x + 4 )} - \frac{2}{33 ( x - 4 )} - \frac{7}{66 ( x + 7 )} + \frac{7}{66 ( x - 7 )}

= 66 \cdot \int \frac{2}{33 ( x + 4 )} - \frac{2}{33 ( x - 4 )} - \frac{7}{66 ( x + 7 )} + \frac{7}{66 ( x - 7 )} 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

= 66 (\int \frac{2}{33 ( x + 4 )} d x - \int \frac{2}{33 ( x - 4 )} d x - \int \frac{7}{66 ( x + 7 )} d x + \int \frac{7}{66 ( x - 7 )} d x)

\int \frac{2}{33 ( x + 4 )} d x = \frac{2}{33} ln ∣ x + 4 ∣

\int \frac{2}{33 ( x - 4 )} d x = \frac{2}{33} ln ∣ x - 4 ∣

\int \frac{7}{66 ( x + 7 )} d x = \frac{7}{66} ln ∣ x + 7 ∣

\int \frac{7}{66 ( x - 7 )} d x = \frac{7}{66} ln ∣ x - 7 ∣

= 66 (\frac{2}{33} ln ∣ x + 4 ∣ - \frac{2}{33} ln ∣ x - 4 ∣ - \frac{7}{66} ln ∣ x + 7 ∣ + \frac{7}{66} ln ∣ x - 7 ∣)

簡素化

66 (\frac{2}{33} ln ∣ x + 4 ∣ - \frac{2}{33} ln ∣ x - 4 ∣ - \frac{7}{66} ln ∣ x + 7 ∣ + \frac{7}{66} ln ∣ x - 7 ∣) : 4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣

= 4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣

定数を解答に追加する

= 4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 7 ln ∣ x + 7 ∣ + 7 ln ∣ x - 7 ∣ + C

グラフ

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