(∫ (x^6+1))/(x^4+1) dx
时间: 2023-02-24 14:02:14 浏览: 88
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