- #1
Hevonen
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Homework Statement
Integrate f(x) = (x^3 + 3x + 12) / (x(x+2)^2)
Homework Equations
The Attempt at a Solution
I know that i should somehow rewrite the polynomial but I do not know how. Please, help me.
The domain of the function is all real numbers except x = 0 and x = -2, since these values would result in division by zero.
The range of the function is all real numbers, since there are no restrictions on the output values.
To find the critical points, set the derivative of the function equal to zero and solve for x. In this case, the derivative is (x^3 + 3x + 12)' = 3x^2 + 3, which gives critical points at x = -1 and x = 1.
To determine the local extrema, evaluate the second derivative of the function at the critical points. If the second derivative is positive, the function has a local minimum at that point. If the second derivative is negative, the function has a local maximum. In this case, the second derivative is 6x, which gives a local minimum at x = -1 and a local maximum at x = 1.
Yes, this function can be integrated using elementary functions. The resulting integral is (x^2 + 6ln(x) - 6ln(x+2)) + C.