integral of 1/(x(x-8)^2)

解答

\int \frac{1}{x ( x - 8 ) ^{2}} d x

\frac{1}{64} ln ∣ x ∣ - \frac{1}{64} ln ∣ x - 8 ∣ - \frac{1}{8 ( x - 8 )} + C

求解步骤

\int \frac{1}{x ( x - 8 ) ^{2}} d x

将

\frac{1}{x ( x - 8 ) ^{2}}

用部份分式展开:

\frac{1}{64 x} - \frac{1}{64 ( x - 8 )} + \frac{1}{8 ( x - 8 ) ^{2}}

= \int \frac{1}{64 x} - \frac{1}{64 ( x - 8 )} + \frac{1}{8 ( x - 8 ) ^{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

= \int \frac{1}{64 x} d x - \int \frac{1}{64 ( x - 8 )} d x + \int \frac{1}{8 ( x - 8 ) ^{2}} d x

\int \frac{1}{64 x} d x = \frac{1}{64} ln ∣ x ∣

\int \frac{1}{64 ( x - 8 )} d x = \frac{1}{64} ln ∣ x - 8 ∣

\int \frac{1}{8 ( x - 8 ) ^{2}} d x = - \frac{1}{8 ( x - 8 )}

= \frac{1}{64} ln ∣ x ∣ - \frac{1}{64} ln ∣ x - 8 ∣ - \frac{1}{8 ( x - 8 )}

解答补常数

= \frac{1}{64} ln ∣ x ∣ - \frac{1}{64} ln ∣ x - 8 ∣ - \frac{1}{8 ( x - 8 )} + C

\frac{d}{d x} (π x^{2} y)

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