integral of (3x^2+1)/((x^3+x)^4)

解答

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

\frac{1}{x} - \frac{1}{3 x ^{3}} + \frac{5}{8} arctan (x) - \frac{3}{8} \cdot \frac{x}{1 + x ^{2}} - \frac{x}{4 ( x ^{2} + 1 ) ^{2}} - \frac{x}{3 ( x ^{2} + 1 ) ^{\frac{5}{2}} x ^{2} + 1} + \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{48 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1} + C

求解步骤

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

将

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

用部份分式展开:

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

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

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

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

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

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

\int \frac{2}{( x ^{2} + 1 ) ^{4}} d x = - 2 - \frac{x}{6 ( 1 + x ^{2} ) ^{\frac{5}{2}} 1 + x ^{2}} + \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{96 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1}

= - (- \frac{1}{x}) - \frac{1}{3 x ^{3}} + arctan (x) - (\frac{3}{8} (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{x}{4 ( x ^{2} + 1 ) ^{2}}) - - 2 - \frac{x}{6 ( 1 + x ^{2} ) ^{\frac{5}{2}} 1 + x ^{2}} + \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{96 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1}

化简

- (- \frac{1}{x}) - \frac{1}{3 x ^{3}} + arctan (x) - (\frac{3}{8} (arctan (x) + \frac{x}{1 + x ^{2}}) + \frac{x}{4 ( x ^{2} + 1 ) ^{2}}) - - 2 - \frac{x}{6 ( 1 + x ^{2} ) ^{\frac{5}{2}} 1 + x ^{2}} + \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{96 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1} : \frac{1}{x} - \frac{1}{3 x ^{3}} + \frac{5}{8} arctan (x) - \frac{3}{8} \cdot \frac{x}{1 + x ^{2}} - \frac{x}{4 ( x ^{2} + 1 ) ^{2}} - \frac{x}{3 ( x ^{2} + 1 ) ^{\frac{5}{2}} x ^{2} + 1} + \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{48 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1}

= \frac{1}{x} - \frac{1}{3 x ^{3}} + \frac{5}{8} arctan (x) - \frac{3}{8} \cdot \frac{x}{1 + x ^{2}} - \frac{x}{4 ( x ^{2} + 1 ) ^{2}} - \frac{x}{3 ( x ^{2} + 1 ) ^{\frac{5}{2}} x ^{2} + 1} + \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{48 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1}

解答补常数

= \frac{1}{x} - \frac{1}{3 x ^{3}} + \frac{5}{8} arctan (x) - \frac{3}{8} \cdot \frac{x}{1 + x ^{2}} - \frac{x}{4 ( x ^{2} + 1 ) ^{2}} - \frac{x}{3 ( x ^{2} + 1 ) ^{\frac{5}{2}} x ^{2} + 1} + \frac{5 ( - 4 x + 3 ( 2 arccot ( x ) - sin ( 2 arccot ( x ) ) ) ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1 )}{48 ( x ^{2} + 1 ) ^{\frac{3}{2}} x ^{2} + 1} + C

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