integral of (w^2)/((w-6)(w^2-36))

解

\int \frac{w ^{2}}{( w - 6 ) ( w ^{2} - 36 )} d w

\frac{3}{4} ln ∣ w - 6 ∣ - \frac{3}{w - 6} + \frac{1}{4} ln ∣ w + 6 ∣ + C

解答ステップ

\int \frac{w ^{2}}{( w - 6 ) ( w ^{2} - 36 )} d w

以下の部分分数を得る:

\frac{w ^{2}}{( w - 6 ) ( w ^{2} - 36 )} : \frac{3}{4 ( w - 6 )} + \frac{3}{( w - 6 ) ^{2}} + \frac{1}{4 ( w + 6 )}

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

\int \frac{3}{4 ( w - 6 )} d w = \frac{3}{4} ln ∣ w - 6 ∣

\int \frac{3}{( w - 6 ) ^{2}} d w = - \frac{3}{w - 6}

\int \frac{1}{4 ( w + 6 )} d w = \frac{1}{4} ln ∣ w + 6 ∣

= \frac{3}{4} ln ∣ w - 6 ∣ - \frac{3}{w - 6} + \frac{1}{4} ln ∣ w + 6 ∣

定数を解答に追加する

= \frac{3}{4} ln ∣ w - 6 ∣ - \frac{3}{w - 6} + \frac{1}{4} ln ∣ w + 6 ∣ + C