integral of ((x^7+2))/((x^2+x+1)^2)

解答

\int \frac{( x ^{7} + 2 )}{( x ^{2} + x + 1 ) ^{2}} d x

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

求解步骤

\int \frac{x ^{7} + 2}{( x ^{2} + x + 1 ) ^{2}} d x

将

\frac{x ^{7} + 2}{( x ^{2} + x + 1 ) ^{2}}

用部份分式展开:

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

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

\int x^{3} d x = \frac{x ^{4}}{4}

\int 2 x^{2} d x = \frac{2 x ^{3}}{3}

\int x d x = \frac{x ^{2}}{2}

\int 2 d x = 2 x

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

\int \frac{x + 2}{( 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{4}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))

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

解答补常数

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

\int \frac{5 x ^{2} + 2 x + 29}{( x - 3 ) ( x ^{2} + 7 )} d x

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