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

解答

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

\frac{1}{2} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{2 x + 1}{3}) + 23 arctan (\frac{1}{3} (2 x + 1)) + C

求解步骤

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

乘开

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

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

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

\int \frac{3}{x ^{2} + x + 1} d x = 23 arctan (\frac{1}{3} (2 x + 1))

= \frac{1}{2} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{2 x + 1}{3}) + 23 arctan (\frac{1}{3} (2 x + 1))

解答补常数

= \frac{1}{2} ln 4 x^{2} + 4 x + 4 - \frac{1}{3} arctan (\frac{2 x + 1}{3}) + 23 arctan (\frac{1}{3} (2 x + 1)) + C

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

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