integral of 1/((11-10x-x^2)^2)

解

\int \frac{1}{( 11 - 10 x - x ^{2} ) ^{2}} d x

- \frac{1}{864} ln ∣ x - 1 ∣ - \frac{1}{144 ( x - 1 )} + \frac{1}{864} ln ∣ x + 11 ∣ - \frac{1}{144 ( x + 11 )} + C

解答ステップ

\int \frac{1}{( 11 - 10 x - x ^{2} ) ^{2}} d x

以下の部分分数を得る:

\frac{1}{( 11 - 10 x - x ^{2} ) ^{2}} : - \frac{1}{864 ( x - 1 )} + \frac{1}{144 ( x - 1 ) ^{2}} + \frac{1}{864 ( x + 11 )} + \frac{1}{144 ( x + 11 ) ^{2}}

= \int - \frac{1}{864 ( x - 1 )} + \frac{1}{144 ( x - 1 ) ^{2}} + \frac{1}{864 ( x + 11 )} + \frac{1}{144 ( x + 11 ) ^{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}{864 ( x - 1 )} d x + \int \frac{1}{144 ( x - 1 ) ^{2}} d x + \int \frac{1}{864 ( x + 11 )} d x + \int \frac{1}{144 ( x + 11 ) ^{2}} d x

\int \frac{1}{864 ( x - 1 )} d x = \frac{1}{864} ln ∣ x - 1 ∣

\int \frac{1}{144 ( x - 1 ) ^{2}} d x = - \frac{1}{144 ( x - 1 )}

\int \frac{1}{864 ( x + 11 )} d x = \frac{1}{864} ln ∣ x + 11 ∣

\int \frac{1}{144 ( x + 11 ) ^{2}} d x = - \frac{1}{144 ( x + 11 )}

= - \frac{1}{864} ln ∣ x - 1 ∣ - \frac{1}{144 ( x - 1 )} + \frac{1}{864} ln ∣ x + 11 ∣ - \frac{1}{144 ( x + 11 )}

定数を解答に追加する

= - \frac{1}{864} ln ∣ x - 1 ∣ - \frac{1}{144 ( x - 1 )} + \frac{1}{864} ln ∣ x + 11 ∣ - \frac{1}{144 ( x + 11 )} + C