Question

In Exercises \(5-14,\) evaluate the integral. $$\int \frac{8 x-7}{2 x^{2}-x-3} d x$$

Short answer

The value of the integral is \(2*ln|2x+1| - 3*ln|x-3| + C\).

Step-by-step solution

Apply partial fraction decomposition

Rewrite \(\frac{8x-7}{2x^2-x-3}\) in the form of \(A/x-a\) + \(B/x-b\), where A and B are coefficients and a and b are roots of the quadratic equation \(2x^2-x-3 = 0\). Factor the denominator to \(2x^2 - x - 3 = (2x+1)(x-3)\) and set different fractions equal to \(A=(2x+1)\) and \(B=(x-3)\). Use linear system to solve for A and B.

Solve the system of equations

Equating the coefficients on each side to form a system of linear equations, we get \(A - 3A = 8\) and \(-B - AB = -7\). Solving this system gives \(A = 2\) and \(B = -3\).

Substitute A and B back to the fraction

Substitute A and B back into the fractions \(\frac{8x-7}{2x^2-x-3}\) = \(\frac{2}{2x+1}\) - \(\frac{3}{x-3}\).

Perform the Integral

Now, integrate term by term to get \(\int \frac{2}{2x+1} dx - \int \frac{3}{x-3} dx\). This leads to the result \(2*ln|2x+1| - 3*ln|x-3|\).

Write the final result

The final answer is \(2*ln|2x+1| - 3*ln|x-3| + C\), where C is the constant of integration. Here, the |x| denotes the absolute value of x.