Problem 3 - Active Calculus

Question

Calculate the integral \(\int \frac{7 x+3}{x^{2}-3 x+2} d x=\) ________________

Step-by-Step Solution

Answer

2 \ln |x - 1| + 5 \ln |x - 2| + C

Step 1 - Factorize the Denominator

First, factorize the quadratic expression in the denominator: \[ x^2 - 3x + 2 \]This can be written as: \[ (x - 1)(x - 2) \]

Step 2 - Set Up Partial Fraction Decomposition

Express the integrand using partial fractions: \[ \frac{7x + 3}{(x - 1)(x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 2} \]We need to find the values of A and B.

Step 3 - Solve for A and B

Multiply both sides by the denominator \[ (x - 1)(x - 2) \] to get: \[ 7x + 3 = A(x - 2) + B(x - 1) \]Solve for A and B by setting up a system of equations.

Step 4 - System of Equations

Equate the coefficients of like terms from both sides: \[ 7x + 3 = Ax - 2A + Bx - B \]This gives two equations:\[ 7 = A + B \]\[ 3 = -2A - B \]

Step 5 - Solve the System

Solve the system of equations:\[ 7 = A + B \] and \[ 3 = -2A - B \]This yields:\[ A = 2 \] and \[ B = 5 \]

Step 6 - Rewrite the Integral

Substitute A and B back into the partial fractions:\[ \frac{7x + 3}{(x - 1)(x - 2)} = \frac{2}{x - 1} + \frac{5}{x - 2} \]Rewrite the integral as:\[ \int \left( \frac{2}{x - 1} + \frac{5}{x - 2} \right) dx \]

Step 7 - Integrate Term by Term

Compute the integral by integrating each term separately:\[ \int \frac{2}{x - 1} dx + \int \frac{5}{x - 2} dx \]This results in:\[ 2 \ln |x - 1| + 5 \ln |x - 2| + C \]

Key Concepts

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a complex rational expression into simpler fractions that can be easily integrated. This method is especially useful when dealing with integrals that involve polynomial fractions.
To perform partial fraction decomposition, you must first factorize the denominator completely. For example, given the expression: \[ \frac{7x + 3}{x^2 - 3x + 2} \] we first factorize the denominator: \[ x^2 - 3x + 2 = (x - 1)(x - 2) \] Next, express the integrand as a sum of simpler fractions: \[ \frac{7x + 3}{(x - 1)(x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 2} \] Here, A and B are constants that need to be determined. To find A and B, we multiply both sides by the common denominator and simplify, then solve a system of equations. This decomposition simplifies integration.

Solving System of Equations

To determine the constants in partial fraction decomposition, we set up a system of linear equations. For instance, from: \[ \frac{7x + 3}{(x - 1)(x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 2} \] Multiplying both sides by \[ (x - 1)(x - 2) \], we get: \[ 7x + 3 = A(x - 2) + B(x - 1) \] Then, expand and combine like terms: \[ 7x + 3 = Ax - 2A + Bx - B \] This gives a system of equations by equating the coefficients: \[ 7 = A + B \] \[ 3 = -2A - B \] Solving these equations simultaneously, we find the values: \[ A = 2 \] \[ B = 5 \] Once we solve for A and B, we can rewrite the original complex fraction as a sum of simpler fractions.

Integral Computation

Once the partial fractions are determined, the next step is to compute the integral. Given the expression: \[ \frac{7x + 3}{(x - 1)(x - 2)} = \frac{2}{x - 1} + \frac{5}{x - 2} \] we first rewrite the integral in a simpler form: \[ \int \left( \frac{2}{x - 1} + \frac{5}{x - 2} \right) dx \] Now, integrate each term separately: \[ \int \frac{2}{x - 1} \, dx + \int \frac{5}{x - 2} \, dx \] Utilize the fact that the integral of \[ \frac{1}{x - a} \] is \[ \ln |x - a| \]. Thus, we get: \[ 2 \ln |x - 1| + 5 \ln |x - 2| + C \] This result shows that by breaking down the integrand into partial fractions and solving, we can seamlessly find the integral.