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

解答

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

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

求解步骤

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

提出常数:

\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

将

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

用部份分式展开:

\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

使用积分加法定则:

\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 )})

化简

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}

解答补常数

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

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

\int 1 - y^{2} d y

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

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

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