integral of (x^3)/((x^2+16)^3)

解

\int \frac{x ^{3}}{( x ^{2} + 16 ) ^{3}} d x

- \frac{1}{2 ( x ^{2} + 16 )} + \frac{4}{( x ^{2} + 16 ) ^{2}} + C

解答ステップ

\int \frac{x ^{3}}{( x ^{2} + 16 ) ^{3}} d x

以下の部分分数を得る:

\frac{x ^{3}}{( x ^{2} + 16 ) ^{3}} : \frac{x}{( x ^{2} + 16 ) ^{2}} - \frac{16 x}{( x ^{2} + 16 ) ^{3}}

= \int \frac{x}{( x ^{2} + 16 ) ^{2}} - \frac{16 x}{( x ^{2} + 16 ) ^{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{x}{( x ^{2} + 16 ) ^{2}} d x - \int \frac{16 x}{( x ^{2} + 16 ) ^{3}} d x

\int \frac{x}{( x ^{2} + 16 ) ^{2}} d x = - \frac{1}{2 ( x ^{2} + 16 )}

\int \frac{16 x}{( x ^{2} + 16 ) ^{3}} d x = - \frac{4}{( x ^{2} + 16 ) ^{2}}

= - \frac{1}{2 ( x ^{2} + 16 )} - (- \frac{4}{( x ^{2} + 16 ) ^{2}})

簡素化

= - \frac{1}{2 ( x ^{2} + 16 )} + \frac{4}{( x ^{2} + 16 ) ^{2}}

定数を解答に追加する

= - \frac{1}{2 ( x ^{2} + 16 )} + \frac{4}{( x ^{2} + 16 ) ^{2}} + C