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

解答

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

8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3}))) + C

求解步骤

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

乘开

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

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

\int \frac{x}{( x ^{2} + x + 1 ) ^{2}} d x = 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1)))))

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

= 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))

解答补常数

= 8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3}))) + C

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