积分计算器。定积分和不定积分（反导数）的解

计算器使用以下方法对函数进行积分：替换、有理函数和分数、未定义系数、因式分解、线性分数无理性、部分积分、欧拉代换、微分二项式、模积分、积分函数、幂、三角、双曲转换和分组。为了解决定积分，应用了牛顿-莱布尼茨公式和在函数不连续点处寻找极限

积分变量

上限			+∞
下限			-∞

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计算结果的导数

积分

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

计算结果

\frac{ln ( ∣ x + 1 ∣ )}{12} - \frac{ln ( x ^{2} - x + 1 )}{6} + \frac{arctan ( \frac{2 x - 1}{3} )}{3 3} + \frac{1}{6 ( x + 1 )} + \frac{ln ( ∣ x - 1 ∣ )}{4} + C

详细解题过程

计算

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

因式分解

分解分母

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

待定系数法

\frac{x ^{2}}{( x - 1 ) ( x + 1 ) ^{2} ( x ^{2} - x + 1 )} = \frac{A x + B}{x ^{2} - x + 1} + \frac{C}{x + 1} + \frac{D}{( x + 1 ) ^{2}} + \frac{E}{x - 1}

减少到一个共同点

\frac{x ^{2}}{( x - 1 ) ( x + 1 ) ^{2} ( x ^{2} - x + 1 )} = \frac{( A + C + E ) x ^{4} + ( A + B - C + D + E ) x ^{3} + ( - A + B - 2 D ) x ^{2} + ( - A - B + C + 2 D + E ) x + ( - B - C - D + E )}{( x - 1 ) ( x + 1 ) ^{2} ( x ^{2} - x + 1 )}

部分分式分解

x^{0} : x^{1} : x^{2} : x^{3} : x^{4} : ⎩ ⎨ ⎧ - B - C - D + E = 0 - A - B + C + 2 D + E = 0 - A + B - 2 D = 1 A + B - C + D + E = 0 A + C + E = 0 = ⎩ ⎨ ⎧ A = - \frac{1}{3} B = \frac{1}{3} C = \frac{1}{12} D = - \frac{1}{6} E = \frac{1}{4}

\frac{A x + B}{x ^{2} - x + 1} + \frac{C}{x + 1} + \frac{D}{( x + 1 ) ^{2}} + \frac{E}{x - 1}

\int \frac{\frac{1}{3} - \frac{x}{3}}{x ^{2} - x + 1} + \frac{1}{12 ( x + 1 )} - \frac{1}{6 ( x + 1 ) ^{2}} + \frac{1}{4 ( x - 1 )} d x

变换

\int - \frac{x - 1}{3 ( x ^{2} - x + 1 )} + \frac{1}{12 ( x + 1 )} - \frac{1}{6 ( x + 1 ) ^{2}} + \frac{1}{4 ( x - 1 )} d x

线性性

- \frac{1}{3} \int \frac{x - 1}{x ^{2} - x + 1} d x + \frac{1}{12} \int \frac{1}{x + 1} d x - \frac{1}{6} \int \frac{1}{( x + 1 ) ^{2}} d x + \frac{1}{4} \int \frac{1}{x - 1} d x =

= - \frac{1}{3} \cdot (1) + \frac{1}{12} \cdot (4) - \frac{1}{6} \cdot (5) + \frac{1}{4} \cdot (6)

= \frac{ln ( ∣ x + 1 ∣ )}{12} - \frac{ln ( x ^{2} - x + 1 )}{6} + \frac{arctan ( \frac{2 x - 1}{3} )}{3 3} + \frac{1}{6 ( x + 1 )} + \frac{ln ( ∣ x - 1 ∣ )}{4}

计算

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

(1)

对被加数进行分组

x - 1 = \frac{1}{2} (2 x - 1) - \frac{1}{2}

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

= \frac{1}{2} \cdot (2) + (3)

= \frac{ln ( x ^{2} - x + 1 )}{2} - \frac{arctan ( \frac{2 x - 1}{3} )}{3}

计算

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

(2)

换元法

u = x^{2} - x + 1 d u = (2 x - 1) d x

[1]

\int \frac{1}{u} d u

根据积分表

ln (u)

回代

u = x^{2} - x + 1

[1]

ln (x^{2} - x + 1)

计算

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

(3)

分组

配方法

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

换元法

v = x - \frac{1}{2} x = \frac{2 v + 1}{2} d x = d v

[2]

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

换元法

t = \frac{2 v}{3} v = \frac{3 t}{2} d v = \frac{3}{2} d t

[3]

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

变换

取出公因子

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

根据积分表

- \frac{1}{3} \cdot arctan (t) = - \frac{arctan ( t )}{3}

回代

t = \frac{2 v}{3} v = \frac{3 t}{2}

[3]

- \frac{arctan ( \frac{2 v}{3} )}{3}

回代

v = x - \frac{1}{2} x = \frac{2 v + 1}{2}

[2]

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

计算

\int \frac{1}{x + 1} d x

(4)

微分

\int \frac{1}{x + 1} d (x + 1)

积分表

\int \frac{1}{u} d u = ln (∣ u ∣)

ln (∣ x + 1 ∣)

计算

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

(5)

换元法

u = x + 1 x = u - 1 d x = d u

[4]

\int \frac{1}{u ^{2}} d u

幂函数积分

\int u^{n} d u = \frac{u ^{n + 1}}{n + 1}, (n \neq = - 1)

其中

n = - 2

- \frac{1}{u}

回代

u = x + 1 x = u - 1

[4]

- \frac{1}{x + 1}

计算

\int \frac{1}{x - 1} d x

(6)

微分

\int \frac{1}{x - 1} d (x - 1)

积分表

\int \frac{1}{u} d u = ln (∣ u ∣)

ln (∣ x - 1 ∣)

ln (x)

arctan (x)

d x

C

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