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受欢迎的 >

积分 1/((x^3-1)^2)

解答

积分

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

解答

- \frac{2}{9} ln ∣ x - 1 ∣ + \frac{1}{9} ln 4 x^{2} + 4 x + 4 + \frac{2}{9 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{3 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{9 3} sin (2 arctan (\frac{1}{3} (2 x + 1))) - \frac{1}{9 ( x - 1 )} - \frac{1}{6 ( x ^{2} + x + 1 )} - \frac{1}{9 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1)))) + C

求解步骤

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

将

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

用部份分式展开:

- \frac{2}{9 ( x - 1 )} + \frac{1}{9 ( x - 1 ) ^{2}} + \frac{2 x + 3}{9 ( x ^{2} + x + 1 )} + \frac{x + 1}{3 ( x ^{2} + x + 1 ) ^{2}}

= \int - \frac{2}{9 ( x - 1 )} + \frac{1}{9 ( x - 1 ) ^{2}} + \frac{2 x + 3}{9 ( x ^{2} + x + 1 )} + \frac{x + 1}{3 ( x ^{2} + x + 1 ) ^{2}} 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{2}{9 ( x - 1 )} d x + \int \frac{1}{9 ( x - 1 ) ^{2}} d x + \int \frac{2 x + 3}{9 ( x ^{2} + x + 1 )} d x + \int \frac{x + 1}{3 ( x ^{2} + x + 1 ) ^{2}} d x

\int \frac{2}{9 ( x - 1 )} d x = \frac{2}{9} ln ∣ x - 1 ∣

\int \frac{1}{9 ( x - 1 ) ^{2}} d x = - \frac{1}{9 ( x - 1 )}

\int \frac{2 x + 3}{9 ( x ^{2} + x + 1 )} d x = \frac{1}{9} (ln 4 x^{2} + 4 x + 4 - \frac{2}{3} arctan (\frac{2 x + 1}{3}) + 23 arctan (\frac{1}{3} (2 x + 1)))

\int \frac{x + 1}{3 ( x ^{2} + x + 1 ) ^{2}} d x = \frac{1}{3} (8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3}))))

= - \frac{2}{9} ln ∣ x - 1 ∣ - \frac{1}{9 ( x - 1 )} + \frac{1}{9} (ln 4 x^{2} + 4 x + 4 - \frac{2}{3} arctan (\frac{2 x + 1}{3}) + 23 arctan (\frac{1}{3} (2 x + 1))) + \frac{1}{3} (8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3}))))

化简

- \frac{2}{9} ln ∣ x - 1 ∣ - \frac{1}{9 ( x - 1 )} + \frac{1}{9} (ln 4 x^{2} + 4 x + 4 - \frac{2}{3} arctan (\frac{2 x + 1}{3}) + 23 arctan (\frac{1}{3} (2 x + 1))) + \frac{1}{3} (8 (- \frac{1}{4 ( 4 x ^{2} + 4 x + 4 )} - \frac{1}{24 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))) + \frac{2}{3 3} (2 arctan (\frac{2 x + 1}{3}) + sin (2 arctan (\frac{2 x + 1}{3})))) : - \frac{2}{9} ln ∣ x - 1 ∣ + \frac{1}{9} ln 4 x^{2} + 4 x + 4 + \frac{2}{9 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{3 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{9 3} sin (2 arctan (\frac{1}{3} (2 x + 1))) - \frac{1}{9 ( x - 1 )} - \frac{1}{6 ( x ^{2} + x + 1 )} - \frac{1}{9 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))

= - \frac{2}{9} ln ∣ x - 1 ∣ + \frac{1}{9} ln 4 x^{2} + 4 x + 4 + \frac{2}{9 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{3 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{9 3} sin (2 arctan (\frac{1}{3} (2 x + 1))) - \frac{1}{9 ( x - 1 )} - \frac{1}{6 ( x ^{2} + x + 1 )} - \frac{1}{9 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1))))

解答补常数

= - \frac{2}{9} ln ∣ x - 1 ∣ + \frac{1}{9} ln 4 x^{2} + 4 x + 4 + \frac{2}{9 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{3 3} arctan (\frac{1}{3} (2 x + 1)) + \frac{2}{9 3} sin (2 arctan (\frac{1}{3} (2 x + 1))) - \frac{1}{9 ( x - 1 )} - \frac{1}{6 ( x ^{2} + x + 1 )} - \frac{1}{9 3} (2 arctan (\frac{1}{3} (2 x + 1)) + sin (2 arctan (\frac{1}{3} (2 x + 1)))) + C

作图

流行的例子

积分 ((e^x+e^{-x}))/2 积分

\frac{( e ^{x} + e ^{- x} )}{2}

积分 (2x)/((x^2+1)^2)积分

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

积分 y^2x^3 积分

y^{2} x^{3}

积分 1/((x^2-3))积分

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

积分 log_{10}(x^2+y^2)积分

lo g_{10} (x^{2} + y^{2})