integral of 6/(x(x-6)^2)

解答

\int \frac{6}{x ( x - 6 ) ^{2}} d x

\frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 6 ∣ - \frac{1}{x - 6} + C

求解步骤

\int \frac{6}{x ( x - 6 ) ^{2}} d x

提出常数:

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

= 6 \cdot \int \frac{1}{x ( x - 6 ) ^{2}} d x

将

\frac{1}{x ( x - 6 ) ^{2}}

用部份分式展开:

\frac{1}{36 x} - \frac{1}{36 ( x - 6 )} + \frac{1}{6 ( x - 6 ) ^{2}}

= 6 \cdot \int \frac{1}{36 x} - \frac{1}{36 ( x - 6 )} + \frac{1}{6 ( x - 6 ) ^{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

= 6 (\int \frac{1}{36 x} d x - \int \frac{1}{36 ( x - 6 )} d x + \int \frac{1}{6 ( x - 6 ) ^{2}} d x)

\int \frac{1}{36 x} d x = \frac{1}{36} ln ∣ x ∣

\int \frac{1}{36 ( x - 6 )} d x = \frac{1}{36} ln ∣ x - 6 ∣

\int \frac{1}{6 ( x - 6 ) ^{2}} d x = - \frac{1}{6 ( x - 6 )}

= 6 (\frac{1}{36} ln ∣ x ∣ - \frac{1}{36} ln ∣ x - 6 ∣ - \frac{1}{6 ( x - 6 )})

化简

6 (\frac{1}{36} ln ∣ x ∣ - \frac{1}{36} ln ∣ x - 6 ∣ - \frac{1}{6 ( x - 6 )}) : \frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 6 ∣ - \frac{1}{x - 6}

= \frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 6 ∣ - \frac{1}{x - 6}

解答补常数

= \frac{1}{6} ln ∣ x ∣ - \frac{1}{6} ln ∣ x - 6 ∣ - \frac{1}{x - 6} + C

y^{^{''}} + 4 y^{^{'}} + 5 y = 15 x + 3 e^{- x}

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x \to \infty lim (5 e^{- t} i - \frac{5 t}{t + 7} j + 5 arctan (t k))