求∫1/x^3+1的不定积分
先分解因式:
∫ 1/(x³ + 1) dx = ∫ 1/[(x + 1)(x² - x + 1)] dx
= ∫ A/(x + 1) dx + ∫ (Bx + C)/(x² - x + 1) dx
1 = A(x² - x + 1) + (Bx + C)(x + 1) = Ax² - Ax + A + Bx² + Cx + Bx + C
1 = (A + B)x² + (- A + B + C)x + (A + C)
{ A + B = 0
{ - A + B + C = 0
{ A + C = 1
(A + B) - (- A + B + C) = 0 ==> { 2A - C = 0
{A + C = 1
(2A - C) + (A + C) = 1 ==> 3A = 1 ==> A = 1/3
B = - 1/3,C = 2/3
原式 = (1/3)∫ dx/(x + 1) - (1/3)∫ (x - 2)/(x² - x + 1) dx
= (1/3)∫ d(x + 1)/(x + 1) - (1/3)∫ [(2x - 1)/2 - 3/2]/(x² - x + 1) dx
= (1/3)ln(x + 1) - (1/6)∫ d(x² - x + 1)/(x² - x +1) + (1/2)∫ d(x - 1/2)/[(x - 1/2)² + 3/4]
= (1/3)ln(x + 1) - (1/6)ln(x² - x + 1) + (1/2)(2/√3)arctan[(x - 1/2) * 2/√3] + C
= (1/3)ln(x + 1) - (1/6)ln(x² - x + 1) + (1/√3)arctan(2x/√3 - 1/√3) + C
扩展资料
不定积分的公式
1、∫ a dx = ax + C,a和C都是常数
2、∫ x^a dx = [x^(a + 1)]/(a + 1) + C,其中a为常数且 a ≠ -1
3、∫ 1/x dx = ln|x| + C
4、∫ a^x dx = (1/lna)a^x + C,其中a > 0 且 a ≠ 1
5、∫ e^x dx = e^x + C
6、∫ cosx dx = sinx + C
7、∫ sinx dx = - cosx + C
8、∫ cotx dx = ln|sinx| + C = - ln|cscx| + C
9、∫ tanx dx = - ln|cosx| + C = ln|secx| + C
10、∫ secx dx =ln|cot(x/2)| + C
= (1/2)ln|(1 + sinx)/(1 - sinx)| + C
= - ln|secx - tanx| + C
= ln|secx + tanx| + C
根据牛顿-莱布尼茨公式,许多函数的定积分的计算就可以简便地通过求不定积分来进行。这里要注意不定积分与定积分之间的关系:定积分是一个数,而不定积分是一个表达式,它们仅仅是数学上有一个计算关系。
一个函数,可以存在不定积分,而不存在定积分,也可以存在定积分,而没有不定积分。连续函数,一定存在定积分和不定积分;若在有限区间[a,b]上只有有限个间断点且函数有界,则定积分存在;若有跳跃、可去、无穷间断点,则原函数一定不存在,即不定积分一定不存在。
解答过程如下:
先分解因式:
∫ 1/(x³ + 1) dx = ∫ 1/[(x + 1)(x² - x + 1)] dx
= ∫ A/(x + 1) dx + ∫ (Bx + C)/(x² - x + 1) dx
1 = A(x² - x + 1) + (Bx + C)(x + 1) = Ax² - Ax + A + Bx² + Cx + Bx + C
1 = (A + B)x² + (- A + B + C)x + (A + C)
{ A + B = 0
{ - A + B + C = 0
{ A + C = 1
(A + B) - (- A + B + C) = 0 ==> { 2A - C = 0
{A + C = 1
(2A - C) + (A + C) = 1 ==> 3A = 1 ==> A = 1/3
B = - 1/3,C = 2/3
原式 = (1/3)∫ dx/(x + 1) - (1/3)∫ (x - 2)/(x² - x + 1) dx
= (1/3)∫ d(x + 1)/(x + 1) - (1/3)∫ [(2x - 1)/2 - 3/2]/(x² - x + 1) dx
= (1/3)ln(x + 1) - (1/6)∫ d(x² - x + 1)/(x² - x +1) + (1/2)∫ d(x - 1/2)/[(x - 1/2)² + 3/4]
= (1/3)ln(x + 1) - (1/6)ln(x² - x + 1) + (1/2)(2/√3)arctan[(x - 1/2) * 2/√3] + C
= (1/3)ln(x + 1) - (1/6)ln(x² - x + 1) + (1/√3)arctan(2x/√3 - 1/√3) + C
扩展资料
分部积分:
(uv)'=u'v+uv'
得:u'v=(uv)'-uv'
两边积分得:∫ u'v dx=∫ (uv)' dx - ∫ uv' dx
即:∫ u'v dx = uv - ∫ uv' d,这就是分部积分公式
也可简写为:∫ v du = uv - ∫ u dv
常用积分公式:
1)∫0dx=c
2)∫x^udx=(x^(u+1))/(u+1)+c
3)∫1/xdx=ln|x|+c
4)∫a^xdx=(a^x)/lna+c
5)∫e^xdx=e^x+c
6)∫sinxdx=-cosx+c
7)∫cosxdx=sinx+c
8)∫1/(cosx)^2dx=tanx+c
9)∫1/(sinx)^2dx=-cotx+c
10)∫1/√(1-x^2) dx=arcsinx+c
∫ 1/(x³ + 1) dx = ∫ 1/[(x + 1)(x² - x + 1)] dx
= ∫ A/(x + 1) dx + ∫ (Bx + C)/(x² - x + 1) dx
1 = A(x² - x + 1) + (Bx + C)(x + 1) = Ax² - Ax + A + Bx² + Cx + Bx + C
1 = (A + B)x² + (- A + B + C)x + (A + C)
{ A + B = 0
{ - A + B + C = 0
{ A + C = 1
(A + B) - (- A + B + C) = 0 ==> { 2A - C = 0
{A + C = 1
(2A - C) + (A + C) = 1 ==> 3A = 1 ==> A = 1/3
B = - 1/3,C = 2/3
原式 = (1/3)∫ dx/(x + 1) - (1/3)∫ (x - 2)/(x² - x + 1) dx
= (1/3)∫ d(x + 1)/(x + 1) - (1/3)∫ [(2x - 1)/2 - 3/2]/(x² - x + 1) dx
= (1/3)ln(x + 1) - (1/6)∫ d(x² - x + 1)/(x² - x +1) + (1/2)∫ d(x - 1/2)/[(x - 1/2)² + 3/4]
= (1/3)ln(x + 1) - (1/6)ln(x² - x + 1) + (1/2)(2/√3)arctan[(x - 1/2) * 2/√3] + C
= (1/3)ln(x + 1) - (1/6)ln(x² - x + 1) + (1/√3)arctan(2x/√3 - 1/√3) + C
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