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

解答

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

\frac{1}{4} ln ∣ x - 2 ∣ - \frac{1}{8} (ln x^{2} - 2 x + 2 + 2 arctan (x - 1)) + \frac{x ^{4} arctan ( x - 1 ) + x ^{4} sin ( 2 arctan ( x - 1 ) ) - x ^{3} - 4 x ^{3} arctan ( x - 1 ) - 4 x ^{3} sin ( 2 arctan ( x - 1 ) ) + 8 x ^{2} arctan ( x - 1 ) + 4 x ^{2} + 8 x ^{2} sin ( 2 arctan ( x - 1 ) ) - 8 x arctan ( x - 1 ) - 6 x - 8 x sin ( 2 arctan ( x - 1 ) ) + 4 arctan ( x - 1 ) + 4 sin ( 2 arctan ( x - 1 ) ) + 4}{4 ( x ^{2} - 2 x + 2 ) ^{2}} + C

求解步骤

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

将

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

用部份分式展开:

\frac{1}{4 ( x - 2 )} - \frac{x}{4 ( x ^{2} - 2 x + 2 )} + \frac{- x + 2}{2 ( x ^{2} - 2 x + 2 ) ^{2}}

= \int \frac{1}{4 ( x - 2 )} - \frac{x}{4 ( x ^{2} - 2 x + 2 )} + \frac{- x + 2}{2 ( x ^{2} - 2 x + 2 ) ^{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}{4 ( x - 2 )} d x - \int \frac{x}{4 ( x ^{2} - 2 x + 2 )} d x + \int \frac{- x + 2}{2 ( x ^{2} - 2 x + 2 ) ^{2}} d x

\int \frac{1}{4 ( x - 2 )} d x = \frac{1}{4} ln ∣ x - 2 ∣

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

\int \frac{- x + 2}{2 ( x ^{2} - 2 x + 2 ) ^{2}} d x = \frac{\frac{x ^{3} ( x - 1 )}{x ^{2} - 2 x + 2} + x ^{2} arctan ( x - 1 ) - x ^{2} - \frac{x ^{2} ( x - 1 )}{x ^{2} - 2 x + 2} - \frac{2 x ^{2} ( x - 1 )}{x ^{2} - 2 x + 2} + x ^{2} sin ( 2 arctan ( x - 1 ) ) - 2 x arctan ( x - 1 ) + x + \frac{4 x ( x - 1 )}{x ^{2} - 2 x + 2} - 2 x sin ( 2 arctan ( x - 1 ) ) - \frac{2 ( x - 1 )}{x ^{2} - 2 x + 2} + 2 arctan ( x - 1 ) + 2 sin ( 2 arctan ( x - 1 ) ) + 1}{4 ( x ^{2} - 2 x + 2 )}

= \frac{1}{4} ln ∣ x - 2 ∣ - \frac{1}{8} (ln x^{2} - 2 x + 2 + 2 arctan (x - 1)) + \frac{\frac{x ^{3} ( x - 1 )}{x ^{2} - 2 x + 2} + x ^{2} arctan ( x - 1 ) - x ^{2} - \frac{x ^{2} ( x - 1 )}{x ^{2} - 2 x + 2} - \frac{2 x ^{2} ( x - 1 )}{x ^{2} - 2 x + 2} + x ^{2} sin ( 2 arctan ( x - 1 ) ) - 2 x arctan ( x - 1 ) + x + \frac{4 x ( x - 1 )}{x ^{2} - 2 x + 2} - 2 x sin ( 2 arctan ( x - 1 ) ) - \frac{2 ( x - 1 )}{x ^{2} - 2 x + 2} + 2 arctan ( x - 1 ) + 2 sin ( 2 arctan ( x - 1 ) ) + 1}{4 ( x ^{2} - 2 x + 2 )}

\frac{\frac{x ^{3} ( x - 1 )}{x ^{2} - 2 x + 2} + x ^{2} arctan ( x - 1 ) - x ^{2} - \frac{x ^{2} ( x - 1 )}{x ^{2} - 2 x + 2} - \frac{2 x ^{2} ( x - 1 )}{x ^{2} - 2 x + 2} + x ^{2} sin ( 2 arctan ( x - 1 ) ) - 2 x arctan ( x - 1 ) + x + \frac{4 x ( x - 1 )}{x ^{2} - 2 x + 2} - 2 x sin ( 2 arctan ( x - 1 ) ) - \frac{2 ( x - 1 )}{x ^{2} - 2 x + 2} + 2 arctan ( x - 1 ) + 2 sin ( 2 arctan ( x - 1 ) ) + 1}{4 ( x ^{2} - 2 x + 2 )} = \frac{x ^{4} arctan ( x - 1 ) + x ^{4} sin ( 2 arctan ( x - 1 ) ) - x ^{3} - 4 x ^{3} arctan ( x - 1 ) - 4 x ^{3} sin ( 2 arctan ( x - 1 ) ) + 8 x ^{2} arctan ( x - 1 ) + 4 x ^{2} + 8 x ^{2} sin ( 2 arctan ( x - 1 ) ) - 8 x arctan ( x - 1 ) - 6 x - 8 x sin ( 2 arctan ( x - 1 ) ) + 4 arctan ( x - 1 ) + 4 sin ( 2 arctan ( x - 1 ) ) + 4}{4 ( x ^{2} - 2 x + 2 ) ^{2}}

= \frac{1}{4} ln ∣ x - 2 ∣ - \frac{1}{8} (ln x^{2} - 2 x + 2 + 2 arctan (x - 1)) + \frac{x ^{4} arctan ( x - 1 ) + x ^{4} sin ( 2 arctan ( x - 1 ) ) - x ^{3} - 4 x ^{3} arctan ( x - 1 ) - 4 x ^{3} sin ( 2 arctan ( x - 1 ) ) + 8 x ^{2} arctan ( x - 1 ) + 4 x ^{2} + 8 x ^{2} sin ( 2 arctan ( x - 1 ) ) - 8 x arctan ( x - 1 ) - 6 x - 8 x sin ( 2 arctan ( x - 1 ) ) + 4 arctan ( x - 1 ) + 4 sin ( 2 arctan ( x - 1 ) ) + 4}{4 ( x ^{2} - 2 x + 2 ) ^{2}}

解答补常数

= \frac{1}{4} ln ∣ x - 2 ∣ - \frac{1}{8} (ln x^{2} - 2 x + 2 + 2 arctan (x - 1)) + \frac{x ^{4} arctan ( x - 1 ) + x ^{4} sin ( 2 arctan ( x - 1 ) ) - x ^{3} - 4 x ^{3} arctan ( x - 1 ) - 4 x ^{3} sin ( 2 arctan ( x - 1 ) ) + 8 x ^{2} arctan ( x - 1 ) + 4 x ^{2} + 8 x ^{2} sin ( 2 arctan ( x - 1 ) ) - 8 x arctan ( x - 1 ) - 6 x - 8 x sin ( 2 arctan ( x - 1 ) ) + 4 arctan ( x - 1 ) + 4 sin ( 2 arctan ( x - 1 ) ) + 4}{4 ( x ^{2} - 2 x + 2 ) ^{2}} + C

x \to 0 + lim ((3 x)^{s i n (x)})

25 y^{^{''}} - 40 y^{^{'}} + 16 y = 0, y (0) = 2, y^{^{'}} (0) = - 3

\int 5 sin^{3} (x) cos^{7} (x) d x

\int_{1}^{2} ln x^{2} + 1 d x

\frac{d}{d x} (ln (4 x - 1))