integral of (x+3)/((x+1)(x^2+x+3))

解答

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

\frac{2}{3} ln ∣ x + 1 ∣ - \frac{1}{3} ln 4 x^{2} + 4 x + 12 + \frac{8}{3 11} arctan (\frac{1}{11} (2 x + 1)) + C

求解步骤

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

将

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

用部份分式展开:

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

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

\int \frac{2}{3 ( x + 1 )} d x = \frac{2}{3} ln ∣ x + 1 ∣

\int \frac{- 2 x + 3}{3 ( x ^{2} + x + 3 )} d x = \frac{1}{3 11} (- 11 ln 4 x^{2} + 4 x + 12 + 8 arctan (\frac{2 x + 1}{11}))

= \frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{3 11} (- 11 ln 4 x^{2} + 4 x + 12 + 8 arctan (\frac{2 x + 1}{11}))

化简

\frac{2}{3} ln ∣ x + 1 ∣ + \frac{1}{3 11} (- 11 ln 4 x^{2} + 4 x + 12 + 8 arctan (\frac{2 x + 1}{11})) : \frac{2}{3} ln ∣ x + 1 ∣ - \frac{1}{3} ln 4 x^{2} + 4 x + 12 + \frac{8}{3 11} arctan (\frac{1}{11} (2 x + 1))

= \frac{2}{3} ln ∣ x + 1 ∣ - \frac{1}{3} ln 4 x^{2} + 4 x + 12 + \frac{8}{3 11} arctan (\frac{1}{11} (2 x + 1))

解答补常数

= \frac{2}{3} ln ∣ x + 1 ∣ - \frac{1}{3} ln 4 x^{2} + 4 x + 12 + \frac{8}{3 11} arctan (\frac{1}{11} (2 x + 1)) + C

\int cos (1 - 3 x) d x

\int \frac{x ^{4} + x - 3}{x ^{3}} d x

\int x^{2} 12 + x^{3} d x

\int \frac{3 - x}{2} d x

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