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人気のある微分積分 >

integral of 1/((2x+3)(x-1))

解

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

解

- \frac{1}{5} ln ∣ 2 x + 3 ∣ + \frac{1}{5} ln ∣ 2 x - 2 ∣ + C

解答ステップ

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

u置換積分法を適用する

= \int \frac{1}{u ( u - 5 )} d u

以下の部分分数を得る:

\frac{1}{u ( u - 5 )} : - \frac{1}{5 u} + \frac{1}{5 ( u - 5 )}

= \int - \frac{1}{5 u} + \frac{1}{5 ( u - 5 )} d u

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{1}{5 u} d u + \int \frac{1}{5 ( u - 5 )} d u

\int \frac{1}{5 u} d u = \frac{1}{5} ln ∣ u ∣

\int \frac{1}{5 ( u - 5 )} d u = \frac{1}{5} ln ∣ u - 5 ∣

= - \frac{1}{5} ln ∣ u ∣ + \frac{1}{5} ln ∣ u - 5 ∣

代用を戻す

u = 2 x + 3

= - \frac{1}{5} ln ∣ 2 x + 3 ∣ + \frac{1}{5} ln ∣ 2 x + 3 - 5 ∣

簡素化

- \frac{1}{5} ln ∣ 2 x + 3 ∣ + \frac{1}{5} ln ∣ 2 x + 3 - 5 ∣ : - \frac{1}{5} ln ∣ 2 x + 3 ∣ + \frac{1}{5} ln ∣ 2 x - 2 ∣

= - \frac{1}{5} ln ∣ 2 x + 3 ∣ + \frac{1}{5} ln ∣ 2 x - 2 ∣

定数を解答に追加する

= - \frac{1}{5} ln ∣ 2 x + 3 ∣ + \frac{1}{5} ln ∣ 2 x - 2 ∣ + C

グラフ

人気の例

integral of \sqrt[3]{x^2-2x+1}

\int 3 x^{2} - 2 x + 1 d x

integral of 1/(sqrt(225-x^2))

\int \frac{1}{225 - x ^{2}} d x

integral of 3X^2-6X+8

\int 3 X^{2} - 6 X + 8 d X

integral of (ln(x))/(x-xln(x))

\int \frac{ln ( x )}{x - x ln ( x )} d x

integral of sin((pix)/(2y))

\int sin (\frac{π x}{2 y}) d x