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受欢迎的微积分 >

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

解答

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

解答

- \frac{1}{4} ln - 4 x^{4} - 4 x^{2} + 4 + \frac{1}{4 5} (ln \frac{2 x ^{2} + 1}{5} + 1 - ln \frac{2 x ^{2} + 1}{5} - 1) + C

求解步骤

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

使用换元积分法

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{u + 1}{1 - u ^{2} - u} d u

使用换元积分法

= \frac{1}{2} \cdot \int \frac{v}{- v ^{2} + v + 1} d v

对

- v^{2} + v + 1

配方:

- (v - \frac{1}{2})^{2} + \frac{5}{4}

= \frac{1}{2} \cdot \int \frac{v}{- ( v - \frac{1}{2} ) ^{2} + \frac{5}{4}} d v

使用换元积分法

= \frac{1}{2} \cdot \int \frac{2 ( 2 w + 1 )}{- 4 w ^{2} + 5} d w

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot 2 \cdot \int \frac{2 w + 1}{- 4 w ^{2} + 5} d w

乘开

\frac{2 w + 1}{- 4 w ^{2} + 5} : \frac{2 w}{- 4 w ^{2} + 5} + \frac{1}{- 4 w ^{2} + 5}

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

= \frac{1}{2} \cdot 2 (\int \frac{2 w}{- 4 w ^{2} + 5} d w + \int \frac{1}{- 4 w ^{2} + 5} d w)

\int \frac{2 w}{- 4 w ^{2} + 5} d w = - \frac{1}{4} ln - 4 w^{2} + 5

\int \frac{1}{- 4 w ^{2} + 5} d w = \frac{1}{4 5} (ln \frac{2}{5} w + 1 - ln \frac{2}{5} w - 1)

= \frac{1}{2} \cdot 2 (- \frac{1}{4} ln - 4 w^{2} + 5 + \frac{1}{4 5} (ln \frac{2}{5} w + 1 - ln \frac{2}{5} w - 1))

代回

= \frac{1}{2} \cdot 2 (- \frac{1}{4} ln - 4 (x^{2} + 1 - \frac{1}{2})^{2} + 5 + \frac{1}{4 5} (ln \frac{2}{5} (x^{2} + 1 - \frac{1}{2}) + 1 - ln \frac{2}{5} (x^{2} + 1 - \frac{1}{2}) - 1))

化简

\frac{1}{2} \cdot 2 (- \frac{1}{4} ln - 4 (x^{2} + 1 - \frac{1}{2})^{2} + 5 + \frac{1}{4 5} (ln \frac{2}{5} (x^{2} + 1 - \frac{1}{2}) + 1 - ln \frac{2}{5} (x^{2} + 1 - \frac{1}{2}) - 1)) : - \frac{1}{4} ln - 4 x^{4} - 4 x^{2} + 4 + \frac{1}{4 5} (ln \frac{2 x ^{2} + 1}{5} + 1 - ln \frac{2 x ^{2} + 1}{5} - 1)

= - \frac{1}{4} ln - 4 x^{4} - 4 x^{2} + 4 + \frac{1}{4 5} (ln \frac{2 x ^{2} + 1}{5} + 1 - ln \frac{2 x ^{2} + 1}{5} - 1)

解答补常数

= - \frac{1}{4} ln - 4 x^{4} - 4 x^{2} + 4 + \frac{1}{4 5} (ln \frac{2 x ^{2} + 1}{5} + 1 - ln \frac{2 x ^{2} + 1}{5} - 1) + C

作图

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