Evaluate the integral (3x+4)/[(x^2+4)(3-x)]dx
a. (1/2)ln(x^2+4) + ln|3-x| + C
b. (1/2)arctan(x/2) + ln|3-x| + C
c. (1/2)arctan(x/2) - ln|3-x| + C
d. ln|(sqrt(x^2+4)/(3-x))| + C
e. None of the above
To evaluate the given integral, we need to use partial fraction decomposition to break down the rational function into simpler fractions.
1. Factor the denominator: (x^2 + 4)(3 - x) = (x + 2i)(x - 2i)(3 - x) = (x + 2i)(2i - x)(3 - x).
2. Express the given rational function as a sum of partial fractions:
(3x + 4)/[(x + 2i)(2i - x)(3 - x)] = A/(x + 2i) + B/(2i - x) + C/(3 - x).
3. Find the common denominator, which will be the product of the denominators:
(x + 2i)(2i - x)(3 - x).
4. Multiply both sides of the equation by the common denominator to eliminate the denominators:
3x + 4 = A(2i - x)(3 - x) + B(x + 2i)(3 - x) + C(x + 2i)(2i - x).
5. Simplify the resulting equation and group like terms:
3x + 4 = A(6 - 5x + x^2 + 6i - 2ix - xi^2) + B(3x + 6i - x^2 - 3x - 2ix + 2x^2) + C(2i^2 - x^2 + 2ix + 4i - 2xi - 4).
6. Combine like terms and write the resulting equation:
3x + 4 = (4A - 3B + C) + (6A + 6B) + (A - B - 4C)x + (6A + 2B + 4C)i.
7. Equate the coefficients of corresponding terms on both sides of the equation to obtain a system of equations:
4A - 3B + C = 4
6A + 6B = 0
A - B - 4C = 3
6A + 2B + 4C = 0.
8. Solve the system of equations. By solving this system of equations, we find that A = 2/3, B = -2/3, and C = -2/3.
9. Rewrite the original integral using partial fractions:
(3x + 4)/[(x^2 + 4)(3 - x)] = 2/3(x + 2i) + (-2/3)(2i - x) + (-2/3)(3 - x).
10. Integrate each term of the partial fractions separately:
∫[2/3(x + 2i)] dx = (2/3)(1/2)x^2 + (2/3)(2i)x + C1 = (1/3)x^2 + (4/3)ix + C1,
∫[-2/3(2i - x)] dx = (-2/3)(-1/2)x^2 + (-2/3)(-2i)x + C2 = (1/3)x^2 + (4/3)ix + C2,
∫[-2/3(3 - x)] dx = (-2/3)(-x) + (-2/3)(3x/2) + C3 = (2/3)x - (3/2) + C3.
11. Combine the integrated terms:
(1/3)x^2 + (4/3)ix + C1 + (1/3)x^2 + (4/3)ix + C2 + (2/3)x - (3/2) + C3.
12. Simplify the result:
(2/3)x^2 + (8/3)ix + (2/3)x - (3/2) + C1 + C2 + C3.
13. Group the real and imaginary terms:
(2/3)x^2 + (2/3)x - (3/2) + (8/3)ix + C.
The integral of (3x + 4)/[(x^2 + 4)(3 - x)]dx evaluates to:
(2/3)x^2 + (2/3)x - (3/2) + (8/3)ix + C.
This result does not match any of the given options, so the correct answer is e. None of the above.