Integrate. int(x+5)/(x^(2)-2x-3)dx

Factor Denominator: Factor the denominator of the integrand. The denominator x^2 - 2x - 3 can be factored into (x - 3)(x + 1).Partial Fraction Decomposition: Perform partial fraction decomposition. We want to express (x+5)/(x^2-2x-3) as A/(x-3) + B/(x+1).Find A and B: Find the values of A and B. To find A and B, we multiply both sides by the denominator (x^2-2x-3) to get: x + 5 = A(x + 1) + B(x - 3). Now we can solve for A and B by choosing suitable values for x.Solve for B: Choose x = 3 to solve for B. Plugging x = 3 into the equation, we get: 3 + 5 = A(3 + 1) + B(3 - 3), 8 = 4A + 0B, A = 2.Solve for A: Choose x = -1 to solve for A. Plugging x = -1 into the equation, we get: -1 + 5 = A(-1 + 1) + B(-1 - 3), 4 = 0A - 4B, B = -1.Write Decomposition: Write the partial fraction decomposition. Now that we have A = 2 and B = -1, we can write: (x+5)/(x^2-2x-3) = 2/(x-3) - 1/(x+1).Integrate Fractions: Integrate the partial fractions. The integral of 2/(x-3) - 1/(x+1) is: 2 * ln|x-3| - ln|x+1| + C, where C is the constant of integration.Combine Logarithms: Combine the logarithms. Using properties of logarithms, we can combine the two terms: ln|(x-3)^2| - ln|x+1| + C.Simplify Expression: Simplify the expression. The final answer is: ln|(x-3)^2/(x+1)| + C.

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Integrate. $\newline$ $\int \frac{x+5}{x^{2}-2 x-3} d x$

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Evaluate definite integrals using the power rule

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Q. Integrate.

\newline

\int \frac{x+5}{x^{2}-2 x-3} d x

Factor Denominator: Factor the denominator of the integrand. $\newline$ The denominator $x^2 - 2x - 3$ can be factored into $(x - 3)(x + 1)$ .
Partial Fraction Decomposition: Perform partial fraction decomposition. $\newline$ We want to express $(x+5)/(x^2-2x-3)$ as $A/(x-3) + B/(x+1)$ .
Find $A$ and $B$ : Find the values of $A$ and $B$ . To find $A$ and $B$ , we multiply both sides by the denominator $(x^2-2x-3)$ to get: $x + 5 = A(x + 1) + B(x - 3)$ . Now we can solve for $A$ and $B$ by choosing suitable values for $B$ $0$ .
Solve for $B$ : Choose $x = 3$ to solve for $B$ . $\newline$ Plugging $x = 3$ into the equation, we get: $\newline$ $3 + 5 = A(3 + 1) + B(3 - 3)$ , $\newline$ $8 = 4A + 0B$ , $\newline$ $A = 2$ .
Solve for A: Choose $x = -1$ to solve for A. $\newline$ Plugging $x = -1$ into the equation, we get: $\newline$ $-1 + 5 = A(-1 + 1) + B(-1 - 3)$ , $\newline$ $4 = 0A - 4B$ , $\newline$ $B = -1$ .
Write Decomposition: Write the partial fraction decomposition. $\newline$ Now that we have $A = 2$ and $B = -1$ , we can write: $\newline$ $(x+5)/(x^2-2x-3) = 2/(x-3) - 1/(x+1)$ .
Integrate Fractions: Integrate the partial fractions. $\newline$ The integral of $\frac{2}{x-3} - \frac{1}{x+1}$ is: $\newline$ $2 \cdot \ln|x-3| - \ln|x+1| + C$ , where $C$ is the constant of integration.
Combine Logarithms: Combine the logarithms. $\newline$ Using properties of logarithms, we can combine the two terms: $\newline$ $\ln|\$x-3$ ^ $2$ | - \ln|x+ $1$ | + C\).
Simplify Expression: Simplify the expression. $\newline$ The final answer is: $\newline$ $\ln\left|\frac{(x-3)^2}{(x+1)}\right| + C$ .