integral of (12x^3)/((4x^2+1)(x-2)^2)

解答

\int \frac{12 x ^{3}}{( 4 x ^{2} + 1 ) ( x - 2 ) ^{2}} d x

12 (\frac{1}{289} (- \frac{15}{8} ln 4 x^{2} + 1 + 2 arctan (2 x)) + \frac{76}{289} ln ∣ x - 2 ∣ - \frac{8}{17 ( x - 2 )}) + C

求解步骤

\int \frac{12 x ^{3}}{( 4 x ^{2} + 1 ) ( x - 2 ) ^{2}} d x

提出常数:

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

= 12 \cdot \int \frac{x ^{3}}{( 4 x ^{2} + 1 ) ( x - 2 ) ^{2}} d x

将

\frac{x ^{3}}{( 4 x ^{2} + 1 ) ( x - 2 ) ^{2}}

用部份分式展开:

\frac{- 15 x + 4}{289 ( 4 x ^{2} + 1 )} + \frac{76}{289 ( x - 2 )} + \frac{8}{17 ( x - 2 ) ^{2}}

= 12 \cdot \int \frac{- 15 x + 4}{289 ( 4 x ^{2} + 1 )} + \frac{76}{289 ( x - 2 )} + \frac{8}{17 ( 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

= 12 (\int \frac{- 15 x + 4}{289 ( 4 x ^{2} + 1 )} d x + \int \frac{76}{289 ( x - 2 )} d x + \int \frac{8}{17 ( x - 2 ) ^{2}} d x)

\int \frac{- 15 x + 4}{289 ( 4 x ^{2} + 1 )} d x = \frac{1}{289} (- \frac{15}{8} ln 4 x^{2} + 1 + 2 arctan (2 x))

\int \frac{76}{289 ( x - 2 )} d x = \frac{76}{289} ln ∣ x - 2 ∣

\int \frac{8}{17 ( x - 2 ) ^{2}} d x = - \frac{8}{17 ( x - 2 )}

= 12 (\frac{1}{289} (- \frac{15}{8} ln 4 x^{2} + 1 + 2 arctan (2 x)) + \frac{76}{289} ln ∣ x - 2 ∣ - \frac{8}{17 ( x - 2 )})

解答补常数

= 12 (\frac{1}{289} (- \frac{15}{8} ln 4 x^{2} + 1 + 2 arctan (2 x)) + \frac{76}{289} ln ∣ x - 2 ∣ - \frac{8}{17 ( x - 2 )}) + C

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