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人気のある微分積分 >

integral of (48x^2)/((x-15)(x+5)^2)

解

\int \frac{48 x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} d x

解

48 (\frac{9}{16} ln ∣ x - 15 ∣ + \frac{7}{16} ln ∣ x + 5 ∣ + \frac{5}{4 ( x + 5 )}) + C

解答ステップ

\int \frac{48 x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 48 \cdot \int \frac{x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} d x

以下の部分分数を得る:

\frac{x ^{2}}{( x - 15 ) ( x + 5 ) ^{2}} : \frac{9}{16 ( x - 15 )} + \frac{7}{16 ( x + 5 )} - \frac{5}{4 ( x + 5 ) ^{2}}

= 48 \cdot \int \frac{9}{16 ( x - 15 )} + \frac{7}{16 ( x + 5 )} - \frac{5}{4 ( x + 5 ) ^{2}} 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

= 48 (\int \frac{9}{16 ( x - 15 )} d x + \int \frac{7}{16 ( x + 5 )} d x - \int \frac{5}{4 ( x + 5 ) ^{2}} d x)

\int \frac{9}{16 ( x - 15 )} d x = \frac{9}{16} ln ∣ x - 15 ∣

\int \frac{7}{16 ( x + 5 )} d x = \frac{7}{16} ln ∣ x + 5 ∣

\int \frac{5}{4 ( x + 5 ) ^{2}} d x = - \frac{5}{4 ( x + 5 )}

= 48 (\frac{9}{16} ln ∣ x - 15 ∣ + \frac{7}{16} ln ∣ x + 5 ∣ - (- \frac{5}{4 ( x + 5 )}))

簡素化

= 48 (\frac{9}{16} ln ∣ x - 15 ∣ + \frac{7}{16} ln ∣ x + 5 ∣ + \frac{5}{4 ( x + 5 )})

定数を解答に追加する

= 48 (\frac{9}{16} ln ∣ x - 15 ∣ + \frac{7}{16} ln ∣ x + 5 ∣ + \frac{5}{4 ( x + 5 )}) + C

グラフ

人気の例

\frac{d}{d t} (- 6 t)

inverselaplace (2x)/(x^2+2x)+x+5

in v erse l a pl a ce \frac{2 x}{x ^{2} + 2 x} + x + 5

y^{''}-14y^'+48y=0

y^{^{''}} - 14 y^{^{'}} + 48 y = 0

integral of sqrt(x+10)

\int x + 10 d x

integral from 0 to pi/2 of cos(5t)cos(10t)

\int_{0}^{\frac{π}{2}} cos (5 t) cos (10 t) d t