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人気のある微分積分 >

integral of 5/((x-1)(x+4))

解

\int \frac{5}{( x - 1 ) ( x + 4 )} d x

解

ln ∣ x - 1 ∣ - ln ∣ x + 4 ∣ + C

解答ステップ

\int \frac{5}{( x - 1 ) ( x + 4 )} d x

定数を除く:

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

= 5 \cdot \int \frac{1}{( x - 1 ) ( x + 4 )} d x

以下の部分分数を得る:

\frac{1}{( x - 1 ) ( x + 4 )} : \frac{1}{5 ( x - 1 )} - \frac{1}{5 ( x + 4 )}

= 5 \cdot \int \frac{1}{5 ( x - 1 )} - \frac{1}{5 ( x + 4 )} 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

= 5 (\int \frac{1}{5 ( x - 1 )} d x - \int \frac{1}{5 ( x + 4 )} d x)

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

\int \frac{1}{5 ( x + 4 )} d x = \frac{1}{5} ln ∣ x + 4 ∣

= 5 (\frac{1}{5} ln ∣ x - 1 ∣ - \frac{1}{5} ln ∣ x + 4 ∣)

簡素化

5 (\frac{1}{5} ln ∣ x - 1 ∣ - \frac{1}{5} ln ∣ x + 4 ∣) : ln ∣ x - 1 ∣ - ln ∣ x + 4 ∣

= ln ∣ x - 1 ∣ - ln ∣ x + 4 ∣

定数を解答に追加する

= ln ∣ x - 1 ∣ - ln ∣ x + 4 ∣ + C

グラフ

人気の例

integral of (cos(2x))/((1+sin(2x))^4)

\int \frac{cos ( 2 x )}{( 1 + sin ( 2 x ) ) ^{4}} d x

integral of sin^8(3z)cos^5(3z)

\int sin^{8} (3 z) cos^{5} (3 z) d z

integral of (1-x^2)^4

\int (1 - x^{2})^{4} d x

integral of sin(pi)

\int sin (π) d x

integral of (5x)/((x+5)^3)

\int \frac{5 x}{( x + 5 ) ^{3}} d x