Integrate (2x)/(4-3x-x^2) dx

Steps to Solve

1. Set up the integral We are tasked with computing the integral of the function. Set this up as: $$ I = \int \frac{2x}{4 - 3x - x^2} , dx $$

2. Factor the denominator To simplify the integration process, we first factor the denominator $4 - 3x - x^2$. Rearranging gives us: $$ -x^2 - 3x + 4 $$ We can factor this to: $$ -(x^2 + 3x - 4) = -(x-1)(x+4) $$

Thus, the integral can be rewritten as: $$ I = \int \frac{2x}{-(x - 1)(x + 4)} , dx = -2\int \frac{x}{(x - 1)(x + 4)} , dx $$

3. Partial fraction decomposition Next, express the integrand using partial fractions: $$ \frac{x}{(x - 1)(x + 4)} = \frac{A}{x - 1} + \frac{B}{x + 4} $$ Multiplying through by the denominator gives: $$ x = A(x + 4) + B(x - 1) $$

4. Solve for coefficients A and B Expanding and combining like terms leads to: $$ x = (A + B)x + (4A - B) $$ Setting up the system of equations:

5. $A + B = 1$

6. $4A - B = 0$

From the first equation, we can express $B$ as $B = 1 - A$. Substituting into the second equation gives: $$ 4A - (1 - A) = 0 $$ $$ 5A - 1 = 0 \implies A = \frac{1}{5} \implies B = \frac{4}{5} $$

5. Rewrite the integral using partial fractions Thus, we have: $$ \frac{x}{(x - 1)(x + 4)} = \frac{1/5}{x - 1} + \frac{4/5}{x + 4} $$ Now, substitute back into the integral: $$ I = -2 \left( \int \frac{1/5}{x - 1} , dx + \int \frac{4/5}{x + 4} , dx \right) $$

6. Integrate each term Integrating gives: $$ I = -2 \left( \frac{1}{5} \ln |x - 1| + \frac{4}{5} \ln |x + 4| \right) + C $$ Simplifying, we find: $$ I = -\frac{2}{5} \ln |x - 1| - \frac{8}{5} \ln |x + 4| + C $$