这道题怎么解,∫ (x^6+1))/(x^4+1) dx
时间: 2023-02-24 13:10:01 浏览: 106
这道题可以使用分式分解法解决。
首先,将分子分解为x^6和1,将分母分解为x^4和1,并将分数改写为两个分式相除的形式。然后,可以使用分式分解法分别对x^6和x^4求导,并用这些导数代替x^6和x^4。
最后,可以使用把分式的形式转化为另一种形式,以使用标准的积分方法求积分。
具体如下:
1. 将分数改写为两个分式相除的形式:
(x^6)/(x^4) ÷ (1)/(1)
2. 分别对x^6和x^4求导:
d(x^6)/dx = 6x^5
d(x^4)/dx = 4x^3
3. 代替x^6和x^4:
(6x^5)/(4x^3) ÷ (1)/(1)
4. 把分式的形式转化为另一种形式:
(6/4) x^2 ÷ (1)/(1) = (3/2)x^2
5. 对(3/2)x^2求积分:
∫(3/2)x^2 dx = (3/2)x^3/3 + C
其中C是常数,用于描述不确定的积分。
因此,最终的答案为:
∫ (x^6)/(x^4 + 1) dx = (3/2)x^3/3 + C
相关问题
能帮我做做这道题吗∫ (x^6+1)/(x^4+1) dx
是的,我可以帮助你解决这道题目。
这道题可以使用分数分式变形和常用积分法来解决。首先,我们将分数分式分成两个分式:
∫ (x^6)/(x^4 + 1) dx = ∫ x^6 dx / ∫ (x^4 + 1) dx - ∫ (x^6)(-1/(x^4 + 1)^2) dx / ∫ (x^4 + 1) dx
第一个分式很容易求解,可以得到:
∫ x^6 dx = x^7/7 + C_1
关于第二个分式,我们可以使用积分法:拆分分母,得到:
∫ (x^4 + 1) dx = ∫ x^4 dx + ∫ 1 dx
= (x^5)/5 + x + C_2
现在,我们可以将上面的结果代入原式,得到:
∫ (x^6)/(x^4 + 1) dx = (x^7/7 + C_1) / ((x^5)/5 + x + C_2) - ∫ (x^6)(-1/(x^4 + 1)^2) dx / ((x^5)/5 + x + C_2)
这是一个可以求解的式子,但是第二个分式需要进一步解决,这可以使用反正切函数和正切函数进行处理。
最终的结果是:
∫ (x^6)/(x^4 + 1) dx = (x^7/7 + C_1) / ((x^5)/5 + x + C_2) - (1/5)tan^-1(x) + C_3
常数C_1,C_2,C_3取决于你的初始条件。
(∫ (x^6+1))/(x^4+1) dx
The given integral is an improper integral and can be solved using partial fraction decomposition.
We start by rewriting the integrand as a sum of partial fractions:
(x^6 + 1)/(x^4 + 1) = A/x^2 + B/x^2 + C/x^2 + D/x^2 + E/x^2 + F/(x^2 + 1)
where A, B, C, D, E, and F are constants that need to be determined.
We can determine these constants by equating the numerator and denominator of the partial fraction representation to the original integrand and solving for the constants.
This gives us:
x^6 + 1 = A x^4 + B x^2 + C + D/x^2 + E/x^4 + F x^2 / (x^2 + 1)
By setting x = 0, we find that C = -1.
By setting x = ∞, we find that A = 0, E = 0, and F = 1.
Finally, by taking the derivative of both sides with respect to x, we find that:
6x^5 = 4Ax^3 + 2Bx + D/x^3 - 4E/x^5 + 2Fx / (x^2 + 1)
By setting x = 1, we find that:
6 = 4A + 2B + D - 4E + 2F
By substituting the values we have already determined for A, C, E, and F, we find that:
6 = 2B + D + 2
Thus, B = 2 and D = 2.
So the partial fraction decomposition is:
(x^6 + 1)/(x^4 + 1) = (2/x^2) + (2/x^2) - 1/(x^2 + 1)
We can now integrate each term of the partial fraction decomposition separately. The antiderivative of each term can be found using basic integrals:
∫ (2/x^2) dx = -2 * ln |x| + C
∫ -1/(x^2 + 1) dx = -arctan x + C
Thus, the antiderivative of the original integrand is:
∫ (x^6 + 1)/(x^4 + 1) dx = -2 * ln |x| - arctan x + C
相关推荐
![rar](https://img-home.csdnimg.cn/images/20210720083606.png)
![zip](https://img-home.csdnimg.cn/images/20210720083736.png)
![pdf](https://img-home.csdnimg.cn/images/20210720083512.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)
![](https://csdnimg.cn/download_wenku/file_type_ask_c1.png)