integral of 1/((x^2-81)^2)

解

\int \frac{1}{( x ^{2} - 81 ) ^{2}} d x

\frac{1}{2916} ln ∣ x + 9 ∣ - \frac{1}{324 ( x + 9 )} - \frac{1}{2916} ln ∣ x - 9 ∣ - \frac{1}{324 ( x - 9 )} + C

解答ステップ

\int \frac{1}{( x ^{2} - 81 ) ^{2}} d x

以下の部分分数を得る:

\frac{1}{( x ^{2} - 81 ) ^{2}} : \frac{1}{2916 ( x + 9 )} + \frac{1}{324 ( x + 9 ) ^{2}} - \frac{1}{2916 ( x - 9 )} + \frac{1}{324 ( x - 9 ) ^{2}}

= \int \frac{1}{2916 ( x + 9 )} + \frac{1}{324 ( x + 9 ) ^{2}} - \frac{1}{2916 ( x - 9 )} + \frac{1}{324 ( x - 9 ) ^{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

= \int \frac{1}{2916 ( x + 9 )} d x + \int \frac{1}{324 ( x + 9 ) ^{2}} d x - \int \frac{1}{2916 ( x - 9 )} d x + \int \frac{1}{324 ( x - 9 ) ^{2}} d x

\int \frac{1}{2916 ( x + 9 )} d x = \frac{1}{2916} ln ∣ x + 9 ∣

\int \frac{1}{324 ( x + 9 ) ^{2}} d x = - \frac{1}{324 ( x + 9 )}

\int \frac{1}{2916 ( x - 9 )} d x = \frac{1}{2916} ln ∣ x - 9 ∣

\int \frac{1}{324 ( x - 9 ) ^{2}} d x = - \frac{1}{324 ( x - 9 )}

= \frac{1}{2916} ln ∣ x + 9 ∣ - \frac{1}{324 ( x + 9 )} - \frac{1}{2916} ln ∣ x - 9 ∣ - \frac{1}{324 ( x - 9 )}

定数を解答に追加する

= \frac{1}{2916} ln ∣ x + 9 ∣ - \frac{1}{324 ( x + 9 )} - \frac{1}{2916} ln ∣ x - 9 ∣ - \frac{1}{324 ( x - 9 )} + C