integral of (8w+w^2)/((w-7)(w^2+16))

解

\int \frac{8 w + w ^{2}}{( w - 7 ) ( w ^{2} + 16 )} d w

\frac{21}{13} ln ∣ w - 7 ∣ + \frac{1}{13} (- 4 ln w^{2} + 16 + 12 arctan (\frac{w}{4})) + C

解答ステップ

\int \frac{8 w + w ^{2}}{( w - 7 ) ( w ^{2} + 16 )} d w

以下の部分分数を得る:

\frac{8 w + w ^{2}}{( w - 7 ) ( w ^{2} + 16 )} : \frac{21}{13 ( w - 7 )} + \frac{- 8 w + 48}{13 ( w ^{2} + 16 )}

= \int \frac{21}{13 ( w - 7 )} + \frac{- 8 w + 48}{13 ( w ^{2} + 16 )} d w

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{21}{13 ( w - 7 )} d w + \int \frac{- 8 w + 48}{13 ( w ^{2} + 16 )} d w

\int \frac{21}{13 ( w - 7 )} d w = \frac{21}{13} ln ∣ w - 7 ∣

\int \frac{- 8 w + 48}{13 ( w ^{2} + 16 )} d w = \frac{1}{13} (- 4 ln w^{2} + 16 + 12 arctan (\frac{w}{4}))

= \frac{21}{13} ln ∣ w - 7 ∣ + \frac{1}{13} (- 4 ln w^{2} + 16 + 12 arctan (\frac{w}{4}))

定数を解答に追加する

= \frac{21}{13} ln ∣ w - 7 ∣ + \frac{1}{13} (- 4 ln w^{2} + 16 + 12 arctan (\frac{w}{4})) + C