integral of (3x-2)/(x^2+8x+25)

解答

\int \frac{3 x - 2}{x ^{2} + 8 x + 25} d x

\frac{3}{2} ln x^{2} + 8 x + 25 - \frac{14}{3} arctan (\frac{1}{3} (x + 4)) + C

求解步骤

\int \frac{3 x - 2}{x ^{2} + 8 x + 25} d x

乘开

\frac{3 x - 2}{x ^{2} + 8 x + 25} : \frac{3 x}{x ^{2} + 8 x + 25} - \frac{2}{x ^{2} + 8 x + 25}

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{3 x}{x ^{2} + 8 x + 25} d x - \int \frac{2}{x ^{2} + 8 x + 25} d x

\int \frac{3 x}{x ^{2} + 8 x + 25} d x = \frac{3}{2} ln x^{2} + 8 x + 25 - 4 arctan (\frac{1}{3} (x + 4))

\int \frac{2}{x ^{2} + 8 x + 25} d x = \frac{2}{3} arctan (\frac{1}{3} (x + 4))

= \frac{3}{2} ln x^{2} + 8 x + 25 - 4 arctan (\frac{1}{3} (x + 4)) - \frac{2}{3} arctan (\frac{1}{3} (x + 4))

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