integral of (x^2)/((3x+1)^3)

解

\int \frac{x ^{2}}{( 3 x + 1 ) ^{3}} d x

\frac{1}{27} ln ∣ 3 x + 1 ∣ + \frac{2}{27 ( 3 x + 1 )} - \frac{1}{54 ( 3 x + 1 ) ^{2}} + C

解答ステップ

\int \frac{x ^{2}}{( 3 x + 1 ) ^{3}} d x

以下の部分分数を得る:

\frac{x ^{2}}{( 3 x + 1 ) ^{3}} : \frac{1}{9 ( 3 x + 1 )} - \frac{2}{9 ( 3 x + 1 ) ^{2}} + \frac{1}{9 ( 3 x + 1 ) ^{3}}

= \int \frac{1}{9 ( 3 x + 1 )} - \frac{2}{9 ( 3 x + 1 ) ^{2}} + \frac{1}{9 ( 3 x + 1 ) ^{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{1}{9 ( 3 x + 1 )} d x - \int \frac{2}{9 ( 3 x + 1 ) ^{2}} d x + \int \frac{1}{9 ( 3 x + 1 ) ^{3}} d x

\int \frac{1}{9 ( 3 x + 1 )} d x = \frac{1}{27} ln ∣ 3 x + 1 ∣

\int \frac{2}{9 ( 3 x + 1 ) ^{2}} d x = - \frac{2}{27 ( 3 x + 1 )}

\int \frac{1}{9 ( 3 x + 1 ) ^{3}} d x = - \frac{1}{54 ( 3 x + 1 ) ^{2}}

= \frac{1}{27} ln ∣ 3 x + 1 ∣ - (- \frac{2}{27 ( 3 x + 1 )}) - \frac{1}{54 ( 3 x + 1 ) ^{2}}

簡素化

= \frac{1}{27} ln ∣ 3 x + 1 ∣ + \frac{2}{27 ( 3 x + 1 )} - \frac{1}{54 ( 3 x + 1 ) ^{2}}

定数を解答に追加する

= \frac{1}{27} ln ∣ 3 x + 1 ∣ + \frac{2}{27 ( 3 x + 1 )} - \frac{1}{54 ( 3 x + 1 ) ^{2}} + C