integral of (x^2+2x+3)/((x^2+16)^2)

解答

\int \frac{x ^{2} + 2 x + 3}{( x ^{2} + 16 ) ^{2}} d x

- \frac{x}{2 ( x ^{2} + 16 )} + \frac{3 x}{32 ( x ^{2} + 16 )} + \frac{19}{128} arctan (\frac{x}{4}) - \frac{1}{x ^{2} + 16} + C

求解步骤

\int \frac{x ^{2} + 2 x + 3}{( x ^{2} + 16 ) ^{2}} d x

乘开

\frac{x ^{2} + 2 x + 3}{( x ^{2} + 16 ) ^{2}} : \frac{x ^{2}}{( x ^{2} + 16 ) ^{2}} + \frac{2 x}{( x ^{2} + 16 ) ^{2}} + \frac{3}{( x ^{2} + 16 ) ^{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 ^{2}}{( x ^{2} + 16 ) ^{2}} d x + \int \frac{2 x}{( x ^{2} + 16 ) ^{2}} d x + \int \frac{3}{( x ^{2} + 16 ) ^{2}} d x

\int \frac{x ^{2}}{( x ^{2} + 16 ) ^{2}} d x = \frac{1}{4} (\frac{1}{2} arctan (\frac{x}{4}) - \frac{2 x}{16 + x ^{2}})

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

\int \frac{3}{( x ^{2} + 16 ) ^{2}} d x = \frac{3}{128} (arctan (\frac{x}{4}) + \frac{4 x}{16 + x ^{2}})

= \frac{1}{4} (\frac{1}{2} arctan (\frac{x}{4}) - \frac{2 x}{16 + x ^{2}}) - \frac{1}{x ^{2} + 16} + \frac{3}{128} (arctan (\frac{x}{4}) + \frac{4 x}{16 + x ^{2}})

化简

\frac{1}{4} (\frac{1}{2} arctan (\frac{x}{4}) - \frac{2 x}{16 + x ^{2}}) - \frac{1}{x ^{2} + 16} + \frac{3}{128} (arctan (\frac{x}{4}) + \frac{4 x}{16 + x ^{2}}) : - \frac{x}{2 ( x ^{2} + 16 )} + \frac{3 x}{32 ( x ^{2} + 16 )} + \frac{19}{128} arctan (\frac{x}{4}) - \frac{1}{x ^{2} + 16}

= - \frac{x}{2 ( x ^{2} + 16 )} + \frac{3 x}{32 ( x ^{2} + 16 )} + \frac{19}{128} arctan (\frac{x}{4}) - \frac{1}{x ^{2} + 16}

解答补常数

= - \frac{x}{2 ( x ^{2} + 16 )} + \frac{3 x}{32 ( x ^{2} + 16 )} + \frac{19}{128} arctan (\frac{x}{4}) - \frac{1}{x ^{2} + 16} + C

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