Question

x^2/1+x^4怎么积分

Answer 1

∫ (1 + x²)/(1 + x⁴) dx，上下除以x²

= ∫ (1/x² + 1)/(1/x² + x²) dx

= ∫ d(x - 1/x)/[(1/x)² - 2(1/x)(x) + (x)² + 2]，将分子积分后移进dx里，凑微分

= ∫ d(x - 1/x)/[(x - 1/x)² + (√2)²]

根据公式∫ dx/(a² + x²) = (1/a)arctan(x/a)

= (1/√2)arctan[(x - 1/x)/√2] + C

= (1/√2)arctan[x/√2 - 1/(x√2)] + C

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

Answer 2

一楼的结果不对，那个结果求导后为：x²/(x²-1)²

本题解法技巧较高
∫ x²/(1+x^4) dx
=(1/2)∫ (x²-1+x²+1)/(1+x^4) dx
=(1/2)∫ (x²-1)/(1+x^4) dx + (1/2)∫ (x²+1)/(1+x^4) dx
分子分同除以x²
=(1/2)∫ (1-1/x²)/(1/x²+x²) dx + (1/2)∫ (1+1/x²)/(1/x²+x²) dx
分子放到微分之后
=(1/2)∫ 1/(1/x²+x²) d(x+1/x) + (1/2)∫ 1/(1/x²+x²) d(x-1/x)
=(1/2)∫ 1/(1/x²+x²+2-2) d(x+1/x) + (1/2)∫ 1/(1/x²+x²-2+2) d(x-1/x)
=(1/2)∫ 1/[(x+1/x)²-2] d(x+1/x) + (1/2)∫ 1/[(x-1/x)²+2] d(x-1/x)
=(√2/8)ln|(x+1/x-√2)/(x+1/x+√2)| + (√2/4)arctan[(x-1/x)/√2] + C
=(√2/8)ln|(x²+1-√2x)/(x²+1+√2x)| + (√2/4)arctan[(x-1/x)/√2] + C

【数学之美】团队为您解答，若有不懂请追问，如果解决问题请点下面的“选为满意答案”。

Answer 3

令x=tany
∫(x^2/(1+x^4))dx
=∫(tany^2/(1+tany^4))*(1/(cosy)^2)dy
=∫(siny)^2/((siny)^4+(cosy)^4) dy
=∫(1/2)(1-cos2y)/(1-4(siny)^2(cosy)^2) dy
=(1/2)∫(1-cos2y)/(1-(sin2y)^2) dy
=(1/2)∫1/(1-(sin2y)^2) dy - (1/2)∫cos2y/(1-(sin2y)^2) dy
=(1/4)∫(1/(cos2y)^2)d(2y) - (1/4)∫1/((1-sin2y)(1+sin2y)) d(sin2y)
=(1/4)tan2y - (1/8)∫(1/(1-sin2y) + 1/(1+sin2y))d(sin2y)
=(1/4)tan2y - (1/8)ln((1+sin2y)/(1-sin2y)) + C
=(1/4)tan2y - (1/4)ln|(siny+cosy)/(siny-cosy)| + C
=(1/2)tany/(1-(tany)^2) - (1/4)ln|(tany+1)/(tany-1)| + C
=(1/2)x/(1-x^2) - (1/4)ln|(x+1)/(x-1)| + C

Answer 4

x^2/1+x^4积分x^31+x^5