integral of 1/(51-70x-25x^2)

解

\int \frac{1}{51 - 70 x - 25 x ^{2}} d x

- \frac{1}{100} ln ∣ 5 x - 3 ∣ + \frac{1}{100} ln ∣ 5 x + 17 ∣ + C

解答ステップ

\int \frac{1}{51 - 70 x - 25 x ^{2}} d x

以下の部分分数を得る:

\frac{1}{51 - 70 x - 25 x ^{2}} : - \frac{1}{20 ( 5 x - 3 )} + \frac{1}{20 ( 5 x + 17 )}

= \int - \frac{1}{20 ( 5 x - 3 )} + \frac{1}{20 ( 5 x + 17 )} 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}{20 ( 5 x - 3 )} d x + \int \frac{1}{20 ( 5 x + 17 )} d x

\int \frac{1}{20 ( 5 x - 3 )} d x = \frac{1}{100} ln ∣ 5 x - 3 ∣

\int \frac{1}{20 ( 5 x + 17 )} d x = \frac{1}{100} ln ∣ 5 x + 17 ∣

= - \frac{1}{100} ln ∣ 5 x - 3 ∣ + \frac{1}{100} ln ∣ 5 x + 17 ∣

定数を解答に追加する

= - \frac{1}{100} ln ∣ 5 x - 3 ∣ + \frac{1}{100} ln ∣ 5 x + 17 ∣ + C