integral of (3x^2-33x+91)/((x-5)^3)

解

\int \frac{3 x ^{2} - 33 x + 91}{( x - 5 ) ^{3}} d x

3 ln ∣ x - 5 ∣ + \frac{3}{x - 5} - \frac{1}{2 ( x - 5 ) ^{2}} + C

解答ステップ

\int \frac{3 x ^{2} - 33 x + 91}{( x - 5 ) ^{3}} d x

拡張

\frac{3 x ^{2} - 33 x + 91}{( x - 5 ) ^{3}} : \frac{3 x ^{2}}{( x - 5 ) ^{3}} - \frac{33 x}{( x - 5 ) ^{3}} + \frac{91}{( x - 5 ) ^{3}}

= \int \frac{3 x ^{2}}{( x - 5 ) ^{3}} - \frac{33 x}{( x - 5 ) ^{3}} + \frac{91}{( x - 5 ) ^{3}} 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{3 x ^{2}}{( x - 5 ) ^{3}} d x - \int \frac{33 x}{( x - 5 ) ^{3}} d x + \int \frac{91}{( x - 5 ) ^{3}} d x

\int \frac{3 x ^{2}}{( x - 5 ) ^{3}} d x = 3 (ln ∣ x - 5 ∣ - \frac{10}{x - 5} - \frac{25}{2 ( x - 5 ) ^{2}})

\int \frac{33 x}{( x - 5 ) ^{3}} d x = 33 (- \frac{1}{x - 5} - \frac{5}{2 ( x - 5 ) ^{2}})

\int \frac{91}{( x - 5 ) ^{3}} d x = - \frac{91}{2 ( x - 5 ) ^{2}}

= 3 (ln ∣ x - 5 ∣ - \frac{10}{x - 5} - \frac{25}{2 ( x - 5 ) ^{2}}) - 33 (- \frac{1}{x - 5} - \frac{5}{2 ( x - 5 ) ^{2}}) - \frac{91}{2 ( x - 5 ) ^{2}}

簡素化

3 (ln ∣ x - 5 ∣ - \frac{10}{x - 5} - \frac{25}{2 ( x - 5 ) ^{2}}) - 33 (- \frac{1}{x - 5} - \frac{5}{2 ( x - 5 ) ^{2}}) - \frac{91}{2 ( x - 5 ) ^{2}} : 3 ln ∣ x - 5 ∣ + \frac{3}{x - 5} - \frac{1}{2 ( x - 5 ) ^{2}}

= 3 ln ∣ x - 5 ∣ + \frac{3}{x - 5} - \frac{1}{2 ( x - 5 ) ^{2}}

定数を解答に追加する

= 3 ln ∣ x - 5 ∣ + \frac{3}{x - 5} - \frac{1}{2 ( x - 5 ) ^{2}} + C