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人気のある微分積分 >

integral of 4/(x^3-5x^2+7x-3)

解

\int \frac{4}{x ^{3} - 5 x ^{2} + 7 x - 3} d x

解

- ln ∣ x - 1 ∣ + \frac{2}{x - 1} + ln ∣ x - 3 ∣ + C

解答ステップ

\int \frac{4}{x ^{3} - 5 x ^{2} + 7 x - 3} d x

定数を除く:

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

= 4 \cdot \int \frac{1}{x ^{3} - 5 x ^{2} + 7 x - 3} d x

以下の部分分数を得る:

\frac{1}{x ^{3} - 5 x ^{2} + 7 x - 3} : - \frac{1}{4 ( x - 1 )} - \frac{1}{2 ( x - 1 ) ^{2}} + \frac{1}{4 ( x - 3 )}

= 4 \cdot \int - \frac{1}{4 ( x - 1 )} - \frac{1}{2 ( x - 1 ) ^{2}} + \frac{1}{4 ( x - 3 )} 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

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

\int \frac{1}{4 ( x - 1 )} d x = \frac{1}{4} ln ∣ x - 1 ∣

\int \frac{1}{2 ( x - 1 ) ^{2}} d x = - \frac{1}{2 ( x - 1 )}

\int \frac{1}{4 ( x - 3 )} d x = \frac{1}{4} ln ∣ x - 3 ∣

= 4 (- \frac{1}{4} ln ∣ x - 1 ∣ - (- \frac{1}{2 ( x - 1 )}) + \frac{1}{4} ln ∣ x - 3 ∣)

簡素化

4 (- \frac{1}{4} ln ∣ x - 1 ∣ - (- \frac{1}{2 ( x - 1 )}) + \frac{1}{4} ln ∣ x - 3 ∣) : - ln ∣ x - 1 ∣ + \frac{2}{x - 1} + ln ∣ x - 3 ∣

= - ln ∣ x - 1 ∣ + \frac{2}{x - 1} + ln ∣ x - 3 ∣

定数を解答に追加する

= - ln ∣ x - 1 ∣ + \frac{2}{x - 1} + ln ∣ x - 3 ∣ + C

グラフ

人気の例

integral of ln(x)-1

\int ln (x) - 1 d x

面積 3/8 x^2,1,2

a re a \frac{3}{8} x^{2}, 1, 2

勾配 (-9,-6),(3,-9)

s l o p e (- 9, - 6), (3, - 9)

xe^{2y}(dy}{dx}+e^{2y}=\frac{ln(x))/x

x e^{2 y} \frac{d y}{d x} + e^{2 y} = \frac{ln ( x )}{x}

integral of 1/(4u^2+3)

\int \frac{1}{4 u ^{2} + 3} d u