Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.)
(x^2−x+12)/(x^3+3x) dx
Using partial fractions,
(x^2-x+12)/(x^3+3x)
= 4/x - (3x+1)/(x^2+3)
The 4/x integrates easily enough. The rest has to be worked on...
3x/(x^2+3) is just another log. For the rest let
x = ?3 tan?
x^2+3 = 3 sec^2?
dx = ?3 sec^2? d?
1/(x^2+3) dx = 1/?3 d?
Now, putting all that together, you get
http://www.wolframalpha.com/input/?i=integral+(x%5E2-x%2B12)%2F(x%5E3%2B3x)
To evaluate the integral of (x^2 − x + 12)/(x^3 + 3x), we can use a method called partial fraction decomposition. This method involves breaking down the fraction into simpler fractions that we can integrate separately.
Step 1: Factorize the denominator
The denominator x^3 + 3x can be factored as x(x^2 + 3). Therefore, the original fraction can be rewritten as (x^2 − x + 12)/(x(x^2 + 3)).
Step 2: Decompose the fraction
We need to decompose the fraction (x^2 − x + 12)/(x(x^2 + 3)) into simpler fractions. The decomposition will have three terms with unknown constants A, B, and C:
(x^2 − x + 12)/(x(x^2 + 3)) = A/x + B/(x^2 + 3) + Cx/(x^2 + 3)
Step 3: Find the common denominator
To add the fractions, we need to find the common denominator, which is x(x^2 + 3).
(x^2 − x + 12)/(x(x^2 + 3)) = (A(x^2 + 3) + Bx(x) + Cx)/(x(x^2 + 3))
Simplifying the numerator:
(x^2 − x + 12)/(x(x^2 + 3)) = (Ax^2 + 3A + Bx^2 + Cx^2 + Cx)/(x(x^2 + 3))
Step 4: Equate numerators
Since the denominators are the same, we can equate the numerators:
x^2 − x + 12 = Ax^2 + Bx^2 + Cx^2 + 3A + Cx
Grouping like terms:
x^2 − x + 12 = (A + B + C)x^2 + (3A + C)x + 3A
By comparing the coefficients, we can obtain the following system of equations:
A + B + C = 1 (for the x^2 terms)
3A + C = −1 (for the x terms)
3A = 12 (for the constant term)
Solving the system of equations, we find:
A = 4
B = -3
C = 0
Step 5: Rewrite the decomposition
Now we can rewrite the decomposed fraction with the determined constants:
(x^2 − x + 12)/(x(x^2 + 3)) = 4/x - 3/(x^2 + 3)
Step 6: Integrate each term
Integrating each term separately, we get:
∫(4/x) dx = 4ln|x| + C1, where C1 is the constant of integration.
∫(-3/(x^2 + 3)) dx = -√3 arctan(x/√3) + C2, where C2 is the constant of integration.
Step 7: Combine the results
Combining the results, the final integral is:
∫((x^2 − x + 12)/(x^3 + 3x)) dx = 4ln|x| - √3 arctan(x/√3) + C, where C = C1 + C2 is the constant of integration.
To evaluate the integral ∫(x^2 - x + 12)/(x^3 + 3x) dx, we can consider using partial fraction decomposition.
The first step is to factorize the denominator x^3 + 3x. Factoring this expression gives us x(x^2 + 3).
Now let's proceed with the partial fraction decomposition:
(x^2 - x + 12)/(x^3 + 3x) = A/x + (Bx + C)/(x^2 + 3)
To find the values of A, B, and C, we need to cross multiply and equate the coefficients of like terms on both sides of the equation:
(x^2 - x + 12) = A(x^2 + 3) + (Bx + C)x
Expanding the right side and equating coefficients, we get:
x^2 - x + 12 = Ax^2 + 3A + Bx^2 + Cx
Equate the coefficients of the like terms:
1) x^2 terms: 1 = A + B
2) x terms: -1 = C
3) constant terms: 12 = 3A
From equations 1) and 3):
A = 4, B = -3
Substituting the values of A, B, and C into the partial fraction decomposition equation, we have:
(x^2 - x + 12)/(x^3 + 3x) = 4/x - (3x - 1)/(x^2 + 3)
Now, we can integrate each term separately:
∫(4/x - (3x - 1)/(x^2 + 3)) dx
For the first term, we have ∫4/x dx, which integrates to 4ln|x| + C1.
For the second term, we use a substitution u = x^2 + 3. Then, du = 2x dx, which can be rewritten as x dx = 0.5du.
After substituting and simplifying, we have:
∫-(3x - 1)/(x^2 + 3) dx = ∫-(3/2)(1/u) du = -3/2 ln|u| + C2
Substituting back u = x^2 + 3, we get:
= -3/2 ln|x^2 + 3| + C2
Finally, combining both terms, the integral becomes:
∫(x^2 - x + 12)/(x^3 + 3x) dx = 4ln|x| - 3/2 ln|x^2 + 3| + C
So, the evaluated integral is 4ln|x| - 3/2 ln|x^2 + 3| + C, where C is the constant of integration.