Integrate (2x)/(43xx^2) dx
Understand the Problem
The question is asking to compute the integral of the function (2x)/(43xx^2) with respect to x. This involves finding an antiderivative or the area under the curve represented by the function.
Answer
$$ I = \frac{2}{5} \ln x  1  \frac{8}{5} \ln x + 4 + C $$
Answer for screen readers
The integral is: $$ I = \frac{2}{5} \ln x  1  \frac{8}{5} \ln x + 4 + C $$
Steps to Solve

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

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: $$ (x^2 + 3x  4) = (x1)(x+4) $$
Thus, the integral can be rewritten as: $$ I = \int \frac{2x}{(x  1)(x + 4)} , dx = 2\int \frac{x}{(x  1)(x + 4)} , dx $$

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: $$ x = A(x + 4) + B(x  1) $$

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:

$A + B = 1$

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

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: $$ I = 2 \left( \int \frac{1/5}{x  1} , dx + \int \frac{4/5}{x + 4} , dx \right) $$

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: $$ I = \frac{2}{5} \ln x  1  \frac{8}{5} \ln x + 4 + C $$
The integral is: $$ I = \frac{2}{5} \ln x  1  \frac{8}{5} \ln x + 4 + C $$
More Information
This integral involves using techniques such as factoring, partial fraction decomposition, and logarithmic integration. The result specifies the antiderivative of the given function, which can be useful for finding areas and solving related problems.
Tips
 Forgetting to include the constant of integration $C$.
 Incorrectly performing the partial fraction decomposition or failing to solve for the constants properly.
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