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人気のある微分積分 >

integral of (18x^2)/(x^4-41x^2+400)

解

\int \frac{18 x ^{2}}{x ^{4} - 41 x ^{2} + 400} d x

解

4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 5 ln ∣ x + 5 ∣ + 5 ln ∣ x - 5 ∣ + C

解答ステップ

\int \frac{18 x ^{2}}{x ^{4} - 41 x ^{2} + 400} d x

定数を除く:

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

= 18 \cdot \int \frac{x ^{2}}{x ^{4} - 41 x ^{2} + 400} d x

以下の部分分数を得る:

\frac{x ^{2}}{x ^{4} - 41 x ^{2} + 400} : \frac{2}{9 ( x + 4 )} - \frac{2}{9 ( x - 4 )} - \frac{5}{18 ( x + 5 )} + \frac{5}{18 ( x - 5 )}

= 18 \cdot \int \frac{2}{9 ( x + 4 )} - \frac{2}{9 ( x - 4 )} - \frac{5}{18 ( x + 5 )} + \frac{5}{18 ( x - 5 )} 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

= 18 (\int \frac{2}{9 ( x + 4 )} d x - \int \frac{2}{9 ( x - 4 )} d x - \int \frac{5}{18 ( x + 5 )} d x + \int \frac{5}{18 ( x - 5 )} d x)

\int \frac{2}{9 ( x + 4 )} d x = \frac{2}{9} ln ∣ x + 4 ∣

\int \frac{2}{9 ( x - 4 )} d x = \frac{2}{9} ln ∣ x - 4 ∣

\int \frac{5}{18 ( x + 5 )} d x = \frac{5}{18} ln ∣ x + 5 ∣

\int \frac{5}{18 ( x - 5 )} d x = \frac{5}{18} ln ∣ x - 5 ∣

= 18 (\frac{2}{9} ln ∣ x + 4 ∣ - \frac{2}{9} ln ∣ x - 4 ∣ - \frac{5}{18} ln ∣ x + 5 ∣ + \frac{5}{18} ln ∣ x - 5 ∣)

簡素化

18 (\frac{2}{9} ln ∣ x + 4 ∣ - \frac{2}{9} ln ∣ x - 4 ∣ - \frac{5}{18} ln ∣ x + 5 ∣ + \frac{5}{18} ln ∣ x - 5 ∣) : 4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 5 ln ∣ x + 5 ∣ + 5 ln ∣ x - 5 ∣

= 4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 5 ln ∣ x + 5 ∣ + 5 ln ∣ x - 5 ∣

定数を解答に追加する

= 4 ln ∣ x + 4 ∣ - 4 ln ∣ x - 4 ∣ - 5 ln ∣ x + 5 ∣ + 5 ln ∣ x - 5 ∣ + C

グラフ

人気の例

derivative of 4x-x^2-3

\frac{d}{d x} (4 x - x^{2} - 3)

limit as y approaches 15 of sqrt(y+3)

y \to 15 lim (y + 3)

integral of (sin(2e^{-4x}))/(e^{4x)}

\int \frac{sin ( 2 e ^{- 4 x} )}{e ^{4 x}} d x

面積 f(x)=x^3-3x^2+3x,g(x)=x

a re a f (x) = x^{3} - 3 x^{2} + 3 x, g (x) = x

derivative f(3)=2x^2+4x

d er i v a t i v e f (3) = 2 x^{2} + 4 x