integral of ((5x+7))/(x^2+4x+4)

解

\int \frac{( 5 x + 7 )}{x ^{2} + 4 x + 4} d x

5 ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C

解答ステップ

\int \frac{5 x + 7}{x ^{2} + 4 x + 4} d x

拡張

\frac{5 x + 7}{x ^{2} + 4 x + 4} : \frac{5 x}{x ^{2} + 4 x + 4} + \frac{7}{x ^{2} + 4 x + 4}

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{5 x}{x ^{2} + 4 x + 4} d x + \int \frac{7}{x ^{2} + 4 x + 4} d x

\int \frac{5 x}{x ^{2} + 4 x + 4} d x = 5 (ln ∣ x + 2 ∣ + \frac{2}{x + 2})

\int \frac{7}{x ^{2} + 4 x + 4} d x = - \frac{7}{x + 2}

= 5 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{7}{x + 2}

簡素化

5 (ln ∣ x + 2 ∣ + \frac{2}{x + 2}) - \frac{7}{x + 2} : 5 ln ∣ x + 2 ∣ + \frac{3}{x + 2}

= 5 ln ∣ x + 2 ∣ + \frac{3}{x + 2}

定数を解答に追加する

= 5 ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C