integral of (x-2)/(x(x^2-4x+5)^2)

解答

\int \frac{x - 2}{x ( x ^{2} - 4 x + 5 ) ^{2}} d x

- \frac{2}{25} ln ∣ x ∣ + \frac{1}{25} (ln x^{2} - 4 x + 5 - 4 arctan (x - 2)) + \frac{x ^{4} arctan ( x - 2 ) + x ^{3} - 8 x ^{3} arctan ( x - 2 ) + 26 x ^{2} arctan ( x - 2 ) - 8 x ^{2} + 21 x - 40 x arctan ( x - 2 ) + 25 arctan ( x - 2 ) - 20}{10 ( x ^{2} - 4 x + 5 ) ^{2}} + C

求解步骤

\int \frac{x - 2}{x ( x ^{2} - 4 x + 5 ) ^{2}} d x

将

\frac{x - 2}{x ( x ^{2} - 4 x + 5 ) ^{2}}

用部份分式展开:

- \frac{2}{25 x} + \frac{2 x - 8}{25 ( x ^{2} - 4 x + 5 )} + \frac{2 x - 3}{5 ( x ^{2} - 4 x + 5 ) ^{2}}

= \int - \frac{2}{25 x} + \frac{2 x - 8}{25 ( x ^{2} - 4 x + 5 )} + \frac{2 x - 3}{5 ( x ^{2} - 4 x + 5 ) ^{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{2}{25 x} d x + \int \frac{2 x - 8}{25 ( x ^{2} - 4 x + 5 )} d x + \int \frac{2 x - 3}{5 ( x ^{2} - 4 x + 5 ) ^{2}} d x

\int \frac{2}{25 x} d x = \frac{2}{25} ln ∣ x ∣

\int \frac{2 x - 8}{25 ( x ^{2} - 4 x + 5 )} d x = \frac{1}{25} (ln x^{2} - 4 x + 5 - 4 arctan (x - 2))

\int \frac{2 x - 3}{5 ( x ^{2} - 4 x + 5 ) ^{2}} d x = \frac{x ^{4} arctan ( x - 2 ) + x ^{3} - 8 x ^{3} arctan ( x - 2 ) + 26 x ^{2} arctan ( x - 2 ) - 8 x ^{2} + 21 x - 40 x arctan ( x - 2 ) + 25 arctan ( x - 2 ) - 20}{10 ( x ^{2} - 4 x + 5 ) ^{2}}

= - \frac{2}{25} ln ∣ x ∣ + \frac{1}{25} (ln x^{2} - 4 x + 5 - 4 arctan (x - 2)) + \frac{x ^{4} arctan ( x - 2 ) + x ^{3} - 8 x ^{3} arctan ( x - 2 ) + 26 x ^{2} arctan ( x - 2 ) - 8 x ^{2} + 21 x - 40 x arctan ( x - 2 ) + 25 arctan ( x - 2 ) - 20}{10 ( x ^{2} - 4 x + 5 ) ^{2}}

解答补常数

= - \frac{2}{25} ln ∣ x ∣ + \frac{1}{25} (ln x^{2} - 4 x + 5 - 4 arctan (x - 2)) + \frac{x ^{4} arctan ( x - 2 ) + x ^{3} - 8 x ^{3} arctan ( x - 2 ) + 26 x ^{2} arctan ( x - 2 ) - 8 x ^{2} + 21 x - 40 x arctan ( x - 2 ) + 25 arctan ( x - 2 ) - 20}{10 ( x ^{2} - 4 x + 5 ) ^{2}} + C

\int e^{(x + e^{x})} d x

\int \frac{3 x ^{2} + 12 x - 20}{x ^{4} - 8 x ^{2} + 16} d x

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\int \frac{1}{3 + 2 x + x ^{2}} d x