Problem 7
Question
Using Partial Fractions In Exercises \(5-22,\) use partial fractions to find the indefinite integral. $$ \int \frac{5}{x^{2}+3 x-4} d x $$
Step-by-Step Solution
Verified Answer
The indefinite integral of \(\int \frac{5}{x^{2}+3 x-4} dx\) is \(-\ln|x-1| + \ln|(x+4)^2| + C\).
1Step 1: Factorizing the Denominator
First factorize the denominator of the rational function. The denominator can be written as \((x-1)(x+4)\) since -1 and 4 are roots of the equation \(x^{2}+3 x-4 = 0\). So, the original integral can be rewritten as: \(\int \frac{5}{(x-1)(x+4)} dx\).
2Step 2: Decomposition into Partial Fractions
Decompose the expression in the integral into simpler fractions. Assign: \(\frac{5}{(x-1)(x+4)} = \frac{A}{x-1} + \frac{B}{x+4}\). This gives us \(5 = A(x+4) + B(x-1)\). Solving for A and B, letting x = -4 we have A = -1 and letting x = 1 we find B = 2.
3Step 3: Applying the Integral
Substitute the values of A and B back into the integral. This simplifies the original problem to: \(\int \frac{-1}{x-1} dx + \int \frac{2}{x+4} dx\). The integral can now be solved using the property \(\int \frac{a}{x-n} dx = a \ln |x - n|\). Thus: \(-\ln|x-1| + 2\ln|x+4| + C\).
4Step 4: Final Step: Writing the Result in Logarithmic Form
The final result can be written in logarithmic form by using the property of logarithms \(\ln(m^n) = n \ln(m)\) to simplify the expression. Thus, the answer is: \(-\ln|x-1| + \ln|(x+4)^2| + C\).
Key Concepts
Indefinite IntegralFactoringDecompositionLogarithmic Form
Indefinite Integral
An indefinite integral represents the collection of antiderivatives of a function. When you perform integration, you are essentially reversing differentiation.
In our exercise, we are asked to find the indefinite integral of the function \(\frac{5}{x^{2} + 3x - 4}\) with respect to \(x\). This means we need to discover a function whose derivative is \(\frac{5}{x^{2} + 3x - 4}\).
The notation \(\int \frac{5}{x^{2} + 3x - 4} \, dx\) is used for indefinite integrals. The result will include a constant \(C\), because when you differentiate an integral, constants disappear. Thus, any constant could be part of the original function before differentiation.
In our exercise, we are asked to find the indefinite integral of the function \(\frac{5}{x^{2} + 3x - 4}\) with respect to \(x\). This means we need to discover a function whose derivative is \(\frac{5}{x^{2} + 3x - 4}\).
The notation \(\int \frac{5}{x^{2} + 3x - 4} \, dx\) is used for indefinite integrals. The result will include a constant \(C\), because when you differentiate an integral, constants disappear. Thus, any constant could be part of the original function before differentiation.
Factoring
Factoring is a method used to break down complex expressions into simpler products of numbers or expressions. In integrals involving rational functions, factoring often simplifies the work significantly.
In the given exercise, the initial step involves factoring the quadratic in the denominator \(x^2 + 3x - 4\). By solving \(x^2 + 3x - 4 = 0\), we find the roots \(-1\) and \(4\). Therefore, the denominator can be factored as \((x - 1)(x + 4)\).
This step essentially transforms the integral into a form that's suitable for partial fraction decomposition, making it easier to manage subsequent operations.
In the given exercise, the initial step involves factoring the quadratic in the denominator \(x^2 + 3x - 4\). By solving \(x^2 + 3x - 4 = 0\), we find the roots \(-1\) and \(4\). Therefore, the denominator can be factored as \((x - 1)(x + 4)\).
This step essentially transforms the integral into a form that's suitable for partial fraction decomposition, making it easier to manage subsequent operations.
Decomposition
Partial fraction decomposition is a powerful algebraic tool used to convert complex fractions into a sum of simpler ones. It is particularly helpful when integrating rational functions.
For our integral, we re-write \(\frac{5}{(x-1)(x+4)}\) as two separate fractions \(\frac{A}{x-1} + \frac{B}{x+4}\). The goal is to determine values for \(A\) and \(B\).
For our integral, we re-write \(\frac{5}{(x-1)(x+4)}\) as two separate fractions \(\frac{A}{x-1} + \frac{B}{x+4}\). The goal is to determine values for \(A\) and \(B\).
- Multiply through by the common denominator to obtain \(5 = A(x+4) + B(x-1)\).
- Choosing \(x = -4\) simplifies to \(A = -1\).
- Choosing \(x = 1\) simplifies to \(B = 2\).
Logarithmic Form
Once decomposition is complete, the problem becomes much simpler. Each fraction fits into the integral of a basic logarithmic form: \(\int \frac{d/dx}{x-n} \, dx = \ln |x - n|\).
In our exercise, the integral is broken into \(\int \frac{-1}{x-1} \, dx + \int \frac{2}{x+4} \, dx\). Each of these terms can be integrated into:
This expression represents the antiderivative of the original function, giving a neat and complete answer to the problem.
In our exercise, the integral is broken into \(\int \frac{-1}{x-1} \, dx + \int \frac{2}{x+4} \, dx\). Each of these terms can be integrated into:
- \(-\ln|x-1|\) for the first term.
- \(2\ln|x+4|\) for the second term.
This expression represents the antiderivative of the original function, giving a neat and complete answer to the problem.
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