integral of (9x^2+x)/((12x^3+2x^2)^2)

解

\int \frac{9 x ^{2} + x}{( 12 x ^{3} + 2 x ^{2} ) ^{2}} d x

\frac{3}{4 x} - \frac{1}{8 x ^{2}} - \frac{9}{2 ( 6 x + 1 )} + C

解答ステップ

\int \frac{9 x ^{2} + x}{( 12 x ^{3} + 2 x ^{2} ) ^{2}} d x

以下の部分分数を得る:

\frac{9 x ^{2} + x}{( 12 x ^{3} + 2 x ^{2} ) ^{2}} : - \frac{3}{4 x ^{2}} + \frac{1}{4 x ^{3}} + \frac{27}{( 6 x + 1 ) ^{2}}

= \int - \frac{3}{4 x ^{2}} + \frac{1}{4 x ^{3}} + \frac{27}{( 6 x + 1 ) ^{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{3}{4 x ^{2}} d x + \int \frac{1}{4 x ^{3}} d x + \int \frac{27}{( 6 x + 1 ) ^{2}} d x

\int \frac{3}{4 x ^{2}} d x = - \frac{3}{4 x}

\int \frac{1}{4 x ^{3}} d x = - \frac{1}{8 x ^{2}}

\int \frac{27}{( 6 x + 1 ) ^{2}} d x = - \frac{9}{2 ( 6 x + 1 )}

= - (- \frac{3}{4 x}) - \frac{1}{8 x ^{2}} - \frac{9}{2 ( 6 x + 1 )}

簡素化

= \frac{3}{4 x} - \frac{1}{8 x ^{2}} - \frac{9}{2 ( 6 x + 1 )}

定数を解答に追加する

= \frac{3}{4 x} - \frac{1}{8 x ^{2}} - \frac{9}{2 ( 6 x + 1 )} + C