integral of (x+3)/((x^2+2x+5)^2)

解答

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

- \frac{1}{2 ( x ^{2} + 2 x + 5 )} + \frac{1}{8} (arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 1)))) + C

求解步骤

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

乘开

\frac{x + 3}{( x ^{2} + 2 x + 5 ) ^{2}} : \frac{x}{( x ^{2} + 2 x + 5 ) ^{2}} + \frac{3}{( x ^{2} + 2 x + 5 ) ^{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} + 2 x + 5 ) ^{2}} d x + \int \frac{3}{( x ^{2} + 2 x + 5 ) ^{2}} d x

\int \frac{x}{( x ^{2} + 2 x + 5 ) ^{2}} d x = - \frac{1}{2 ( x ^{2} + 2 x + 5 )} - \frac{1}{16} (arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 1))))

\int \frac{3}{( x ^{2} + 2 x + 5 ) ^{2}} d x = \frac{3}{16} (arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 1))))

= - \frac{1}{2 ( x ^{2} + 2 x + 5 )} - \frac{1}{16} (arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 1)))) + \frac{3}{16} (arctan (\frac{1}{2} (x + 1)) + \frac{1}{2} sin (2 arctan (\frac{1}{2} (x + 1))))

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