integral of (4w^2-12w+3w^5-17)/(-2+w^2)

解

\int \frac{4 w ^{2} - 12 w + 3 w ^{5} - 17}{- 2 + w ^{2}} d w

\frac{3 w ^{4}}{4} + 3 w^{2} + 4 w + \frac{9}{2 2} (ln \frac{w}{2} + 1 - ln \frac{w}{2} - 1) + C

解答ステップ

\int \frac{4 w ^{2} - 12 w + 3 w ^{5} - 17}{- 2 + w ^{2}} d w

長除法

\frac{4 w ^{2} - 12 w + 3 w ^{5} - 17}{- 2 + w ^{2}} : 3 w^{3} + 6 w + 4 - \frac{9}{w ^{2} - 2}

= \int 3 w^{3} + 6 w + 4 - \frac{9}{w ^{2} - 2} 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 3 w^{3} d w + \int 6 w d w + \int 4 d w - \int \frac{9}{w ^{2} - 2} d w

\int 3 w^{3} d w = \frac{3 w ^{4}}{4}

\int 6 w d w = 3 w^{2}

\int 4 d w = 4 w

\int \frac{9}{w ^{2} - 2} d w = - \frac{9}{2 2} (ln \frac{w}{2} + 1 - ln \frac{w}{2} - 1)

= \frac{3 w ^{4}}{4} + 3 w^{2} + 4 w - (- \frac{9}{2 2} (ln \frac{w}{2} + 1 - ln \frac{w}{2} - 1))

簡素化

= \frac{3 w ^{4}}{4} + 3 w^{2} + 4 w + \frac{9}{2 2} (ln \frac{w}{2} + 1 - ln \frac{w}{2} - 1)

定数を解答に追加する

= \frac{3 w ^{4}}{4} + 3 w^{2} + 4 w + \frac{9}{2 2} (ln \frac{w}{2} + 1 - ln \frac{w}{2} - 1) + C