Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{5 x+7}{x^{2}+4 x+4} d x\)

Short answer

The integral is \(5 \ln|x+2| - \frac{3}{x+2} + C\).

Step-by-step solution

Factor the denominator

We start by factoring the quadratic polynomial in the denominator. The expression is \(x^2 + 4x + 4\). This can be rewritten as \((x + 2)(x + 2)\) or \((x+2)^2\).

Set up partial fraction decomposition

Since the denominator is \((x+2)^2\), we set up the partial fraction decomposition as \(\frac{A}{x+2} + \frac{B}{(x+2)^2}\). This form is used because the factor \((x+2)\) is repeated.

Clear the denominators

Multiply both sides by \((x+2)^2\) to eliminate the denominators: \[5x + 7 = A(x+2) + B\]

Solve for coefficients A and B

Expand and then simplify the equation: \[5x + 7 = Ax + 2A + B\] Equate coefficients to get the following system of equations:1. \(A = 5\)2. \(2A + B = 7\)Substitute \(A = 5\) into the second equation:\[2(5) + B = 7\]Simplify it:\[10 + B = 7 \implies B = -3\]

Rewrite the integrand

Plug the coefficients back into the partial fraction decomposition to get:\[\frac{5}{x+2} + \frac{-3}{(x+2)^2}\]Replace the original integrand with this decomposition:

Integrate each term separately

Now, integrate each term separately:\[\int \frac{5}{x+2} \, dx + \int \frac{-3}{(x+2)^2} \, dx\]The first integral is a basic logarithmic form:\[5 \ln|x+2| + C_1\]The second integral is a power rule for negative powers:\[\frac{3}{x+2} + C_2\]Combine constants of integration \(C_1\) and \(C_2\) into a single constant \(C\).

Combine the results

Combine the results of the integrations:\[5 \ln|x+2| - \frac{3}{x+2} + C\]

Key concepts

Integration

Integration allows us to find the area under a curve, or the anti-derivative of a function. It is a core concept in calculus and is the reverse process of differentiation. Integrals can be definite, providing a numerical value for the area, or indefinite, returning an expression plus a constant of integration. When dealing with integrals like \( \int \frac{5x+7}{x^{2}+4x+4} dx \), partial fraction decomposition is often required. This method enables us to break down complex rational expressions into simpler ones, which are more easily integrated.Overall, integration provides vital tools for mathematics, physics, and engineering, allowing for the calculation of areas, volumes, and more.

Quadratic Polynomial

A quadratic polynomial is any polynomial of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). These polynomials are especially significant in integration as their factorization plays a crucial role in simplifying the integrals of rational expressions. In our exercise, the quadratic polynomial \( x^2 + 4x + 4 \) was factored as \( (x+2)^2 \). Factorization can simplify expressions into products of linear factors, which can be used in partial fraction decomposition to break integrals down into simpler components for easier integration. Recognizing and working with these quadratic forms is an essential skill in calculus.

Logarithmic Integration

Logarithmic integration is a method used to integrate functions of the form \( \frac{1}{x} \) or similar. The integral of \( \frac{1}{x} \) is \( \ln|x| \), and this process can be extended to functions of the form \( \frac{1}{ax+b} \), which integrate to \( \frac{1}{a} \ln|ax+b| \).In our example, one part of the integral, \( \int \frac{5}{x+2} dx \), fits this form exactly. It integrates to \( 5 \ln|x+2| \). These types of integrals are straightforward once you recognize the pattern, and they frequently appear when dealing with rational expressions and partial fraction decomposition.

Power Rule for Integration

The power rule for integration is another fundamental tool that simplifies many integration problems. This rule states that to integrate a function of the form \( x^n \), you add one to the exponent and divide by this new exponent: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \text{ for } n eq -1 \]In the context of our exercise, we deal with a power rule for negative exponents: \( \int \frac{-3}{(x+2)^2} dx \). This is actually already in the form \( x^n \), where \( n = -2 \). Integrating this gives us \( \frac{3}{x+2} \).By understanding and applying this rule effectively, you can tackle a wide range of integrals featuring different powers of \(x\).