integral of (2x+7)/((2x^2+14x)^4)

解

\int \frac{2 x + 7}{( 2 x ^{2} + 14 x ) ^{4}} d x

- \frac{1}{134456 x} + \frac{1}{38416 x ^{2}} - \frac{1}{16464 x ^{3}} + \frac{1}{134456 ( x + 7 )} + \frac{1}{38416 ( x + 7 ) ^{2}} + \frac{1}{16464 ( x + 7 ) ^{3}} + C

解答ステップ

\int \frac{2 x + 7}{( 2 x ^{2} + 14 x ) ^{4}} d x

以下の部分分数を得る:

\frac{2 x + 7}{( 2 x ^{2} + 14 x ) ^{4}} : \frac{1}{134456 x ^{2}} - \frac{1}{19208 x ^{3}} + \frac{1}{5488 x ^{4}} - \frac{1}{134456 ( x + 7 ) ^{2}} - \frac{1}{19208 ( x + 7 ) ^{3}} - \frac{1}{5488 ( x + 7 ) ^{4}}

= \int \frac{1}{134456 x ^{2}} - \frac{1}{19208 x ^{3}} + \frac{1}{5488 x ^{4}} - \frac{1}{134456 ( x + 7 ) ^{2}} - \frac{1}{19208 ( x + 7 ) ^{3}} - \frac{1}{5488 ( x + 7 ) ^{4}} 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}{134456 x ^{2}} d x - \int \frac{1}{19208 x ^{3}} d x + \int \frac{1}{5488 x ^{4}} d x - \int \frac{1}{134456 ( x + 7 ) ^{2}} d x - \int \frac{1}{19208 ( x + 7 ) ^{3}} d x - \int \frac{1}{5488 ( x + 7 ) ^{4}} d x

\int \frac{1}{134456 x ^{2}} d x = - \frac{1}{134456 x}

\int \frac{1}{19208 x ^{3}} d x = - \frac{1}{38416 x ^{2}}

\int \frac{1}{5488 x ^{4}} d x = - \frac{1}{16464 x ^{3}}

\int \frac{1}{134456 ( x + 7 ) ^{2}} d x = - \frac{1}{134456 ( x + 7 )}

\int \frac{1}{19208 ( x + 7 ) ^{3}} d x = - \frac{1}{38416 ( x + 7 ) ^{2}}

\int \frac{1}{5488 ( x + 7 ) ^{4}} d x = - \frac{1}{16464 ( x + 7 ) ^{3}}

= - \frac{1}{134456 x} - (- \frac{1}{38416 x ^{2}}) - \frac{1}{16464 x ^{3}} - (- \frac{1}{134456 ( x + 7 )}) - (- \frac{1}{38416 ( x + 7 ) ^{2}}) - (- \frac{1}{16464 ( x + 7 ) ^{3}})

簡素化

= - \frac{1}{134456 x} + \frac{1}{38416 x ^{2}} - \frac{1}{16464 x ^{3}} + \frac{1}{134456 ( x + 7 )} + \frac{1}{38416 ( x + 7 ) ^{2}} + \frac{1}{16464 ( x + 7 ) ^{3}}

定数を解答に追加する

= - \frac{1}{134456 x} + \frac{1}{38416 x ^{2}} - \frac{1}{16464 x ^{3}} + \frac{1}{134456 ( x + 7 )} + \frac{1}{38416 ( x + 7 ) ^{2}} + \frac{1}{16464 ( x + 7 ) ^{3}} + C