Final answer:
To find the antiderivative of the given expression, we need to use partial fraction decomposition. The antiderivative of (x^3-4x^2-4x+20)/(x^2-5) is x^2/2 + 4x - 18ln|x^2-5| + 18√5 arctan(x/√5).
Step-by-step explanation:
To find the antiderivative of the given expression, we need to use partial fraction decomposition. The numerator, x^3-4x^2-4x+20, is a cubic function while the denominator, x^2-5, is a quadratic function. The degree of the numerator is greater than the degree of the denominator, so we need to divide the numerator by the denominator to get a proper fraction.
Dividing the numerator by the denominator, we get (x^3-4x^2-4x+20)/(x^2-5) = x + 4 + (-36x+180)/(x^2-5).
Now, we can use partial fraction decomposition for the fraction (-36x+180)/(x^2-5). We can rewrite it as (-36x)/(x^2-5) + 180/(x^2-5).
The antiderivative of x is x^2/2, the antiderivative of 4 is 4x, and the antiderivative of (-36x)/(x^2-5) is -18ln|x^2-5|, where ln is the natural logarithm. The antiderivative of 180/(x^2-5) is 18√5 arctan(x/√5), where arctan is the inverse tangent function.
Therefore, the antiderivative of (x^3-4x^2-4x+20)/(x^2-5) is x^2/2 + 4x - 18ln|x^2-5| + 18√5 arctan(x/√5).+