integrate (x^2-x+12)/(x^3+2x)dx
https://www.integral-calculator.com/
oops, I deleted accidently the wrong answer, oobleck's answer was use of partial fractions, if you type in the url above, and indicate show steps, you will see the expression worked in units of partial fractions. Sorry about deleting the wrong response, oob.
To integrate the given expression,
∫(x^2 - x + 12)/(x^3 + 2x) dx,
we can start by performing partial fraction decomposition. This means breaking down the rational function into simpler fractions.
First, let's factor the denominator:
x^3 + 2x = x(x^2 + 2).
Now, we can write the integrand as the sum of two fractions:
(x^2 - x + 12)/(x^3 + 2x) = A/x + (Bx + C)/(x^2 + 2),
where A, B, and C are constants.
To find these constants, we need to clear the fractions by multiplying both sides of the equation by the common denominator (x)(x^2 + 2):
(x^2 - x + 12) = A*(x^2 + 2) + (Bx + C)*(x).
Expanding the equation, we get:
x^2 - x + 12 = Ax^2 + 2A + Bx^2 + Cx.
Next, we equate the coefficients of like powers of x on both sides of the equation. This gives us a system of equations:
1. Coefficients of x^2:
1 = A + B.
2. Coefficients of x:
-1 = C.
3. Coefficients of the constant term:
12 = 2A.
From equation 3, we find A = 6. Then, from equation 1, we find B = -5. Lastly, from equation 2, we find C = -1.
Substituting these values back into the partial fraction decomposition, we get:
(x^2 - x + 12)/(x^3 + 2x) = 6/x - (5x + 1)/(x^2 + 2).
Now, we can integrate each term separately:
∫(6/x - (5x + 1)/(x^2 + 2)) dx.
The integral of 6/x is simply 6 ln|x| + C, where C is the constant of integration.
For the second term, (5x + 1)/(x^2 + 2), we can use a substitution, u = x^2 + 2:
Let u = x^2 + 2. Then du = 2x dx, or dx = du/(2x).
After substituting, the integral becomes:
∫(5x + 1)/(x^2 + 2) dx = ∫(5x + 1)/u (du/(2x)).
Simplifying, we have:
∫(5x + 1)/(x^2 + 2) dx = ∫(5/(2u)) du.
Now, integrating this new expression, we get:
(5/2) ∫(1/u) du = (5/2) ln|u| + C.
Substituting back u = x^2 + 2, we have:
(5/2) ln|x^2 + 2| + C.
Therefore, the final result of the integral is:
6 ln|x| - (5/2) ln|x^2 + 2| + C,
where C is the constant of integration.