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人気のある微分積分 >

integral of 8/((x-1)(x+7))

解

\int \frac{8}{( x - 1 ) ( x + 7 )} d x

解

ln ∣ x - 1 ∣ - ln ∣ x + 7 ∣ + C

解答ステップ

\int \frac{8}{( x - 1 ) ( x + 7 )} d x

定数を除く:

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

= 8 \cdot \int \frac{1}{( x - 1 ) ( x + 7 )} d x

以下の部分分数を得る:

\frac{1}{( x - 1 ) ( x + 7 )} : \frac{1}{8 ( x - 1 )} - \frac{1}{8 ( x + 7 )}

= 8 \cdot \int \frac{1}{8 ( x - 1 )} - \frac{1}{8 ( x + 7 )} 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

= 8 (\int \frac{1}{8 ( x - 1 )} d x - \int \frac{1}{8 ( x + 7 )} d x)

\int \frac{1}{8 ( x - 1 )} d x = \frac{1}{8} ln ∣ x - 1 ∣

\int \frac{1}{8 ( x + 7 )} d x = \frac{1}{8} ln ∣ x + 7 ∣

= 8 (\frac{1}{8} ln ∣ x - 1 ∣ - \frac{1}{8} ln ∣ x + 7 ∣)

簡素化

8 (\frac{1}{8} ln ∣ x - 1 ∣ - \frac{1}{8} ln ∣ x + 7 ∣) : ln ∣ x - 1 ∣ - ln ∣ x + 7 ∣

= ln ∣ x - 1 ∣ - ln ∣ x + 7 ∣

定数を解答に追加する

= ln ∣ x - 1 ∣ - ln ∣ x + 7 ∣ + C

グラフ

人気の例

integral of (3-x)e^x

\int (3 - x) e^{x} d x

integral of x/((x^2+2x+5)^2)

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

integral of (x^2+1)^3(2x)

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

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

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

integral of 1/(sqrt(7x^2+49))

\int \frac{1}{7 x ^{2} + 49} d x