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人気のある微分積分 >

integral of 7/(x(x-7)^2)

解

\int \frac{7}{x ( x - 7 ) ^{2}} d x

解

\frac{1}{7} ln ∣ x ∣ - \frac{1}{7} ln ∣ x - 7 ∣ - \frac{1}{x - 7} + C

解答ステップ

\int \frac{7}{x ( x - 7 ) ^{2}} d x

定数を除く:

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

= 7 \cdot \int \frac{1}{x ( x - 7 ) ^{2}} d x

以下の部分分数を得る:

\frac{1}{x ( x - 7 ) ^{2}} : \frac{1}{49 x} - \frac{1}{49 ( x - 7 )} + \frac{1}{7 ( x - 7 ) ^{2}}

= 7 \cdot \int \frac{1}{49 x} - \frac{1}{49 ( x - 7 )} + \frac{1}{7 ( x - 7 ) ^{2}} 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

= 7 (\int \frac{1}{49 x} d x - \int \frac{1}{49 ( x - 7 )} d x + \int \frac{1}{7 ( x - 7 ) ^{2}} d x)

\int \frac{1}{49 x} d x = \frac{1}{49} ln ∣ x ∣

\int \frac{1}{49 ( x - 7 )} d x = \frac{1}{49} ln ∣ x - 7 ∣

\int \frac{1}{7 ( x - 7 ) ^{2}} d x = - \frac{1}{7 ( x - 7 )}

= 7 (\frac{1}{49} ln ∣ x ∣ - \frac{1}{49} ln ∣ x - 7 ∣ - \frac{1}{7 ( x - 7 )})

簡素化

7 (\frac{1}{49} ln ∣ x ∣ - \frac{1}{49} ln ∣ x - 7 ∣ - \frac{1}{7 ( x - 7 )}) : \frac{1}{7} ln ∣ x ∣ - \frac{1}{7} ln ∣ x - 7 ∣ - \frac{1}{x - 7}

= \frac{1}{7} ln ∣ x ∣ - \frac{1}{7} ln ∣ x - 7 ∣ - \frac{1}{x - 7}

定数を解答に追加する

= \frac{1}{7} ln ∣ x ∣ - \frac{1}{7} ln ∣ x - 7 ∣ - \frac{1}{x - 7} + C

グラフ

人気の例

derivative of (x-100^3+(x-100)^2+500)

\frac{d}{d x} ((x - 100)^{3} + (x - 100)^{2} + 500)

integral of sqrt(1-y^2)

\int 1 - y^{2} d y

derivative of (x^3+7^{2/3})

\frac{d}{d x} ((x^{3} + 7)^{\frac{2}{3}})

derivative g(x)=sqrt(8-x)

d er i v a t i v e g (x) = 8 - x

tangent f(x)=(5x-1)^{-1/6},\at x=13

t an g e n t f (x) = (5 x - 1)^{- \frac{1}{6}}, at x = 13