Problem 30 - Calculus Single Variable

Question

Use the method of partial fractions to decompose the integrand. Then evaluate the given integral. \(\int \frac{2 x^{4}+15 x^{2}+30}{\left(x^{2}+4\right)\left(x^{2}+3\right)^{2}} d x\)

Step-by-Step Solution

Answer

Decompose into partial fractions and solve integrals; sum results for solution.

Step 1: Set Up Partial Fractions

To find the partial fraction decomposition, assume the integrand can be expressed as: \[ \frac{2x^4 + 15x^2 + 30}{(x^2 + 4)(x^2 + 3)^2} = \frac{Ax + B}{x^2 + 4} + \frac{Cx + D}{x^2 + 3} + \frac{Ex + F}{(x^2 + 3)^2}.\] This assumes a form with linear numerators since the denominator consists of quadratic factors.

Step 2: Combine Terms

Multiply through by the common denominator to eliminate fractions:\[ 2x^4 + 15x^2 + 30 = (Ax + B)(x^2 + 3)^2 + (Cx + D)(x^2 + 4)(x^2 + 3) + (Ex + F)(x^2 + 4). \] Expand and collect like terms from the right-hand side.

Step 3: Expand and Collect Like Terms

Expand each term from the previous step:- \((x^2 + 3)^2 = x^4 + 6x^2 + 9\).- Multiply \((Ax + B)\) by \(x^4 + 6x^2 + 9\).- Multiply \((Cx + D)\) by \(x^4 + 7x^2 + 12\).- Multiply \((Ex + F)\) by \(x^2 + 4\).Equating coefficients from the expansion on the right to the original polynomial on the left will allow us to solve for \(A, B, C, D, E, F\).

Step 4: Solve the System of Equations

Compare coefficients of like powers of \(x\) from both sides to form a system of equations, then solve for \(A, B, C, D, E, F\). This leads to a series of linear equations that can be solved algebraically. Using computational tools can ease this process.

Step 5: Integrate Each Part

Once the partial fraction decomposition is determined, integrate each term separately:- \( \int \frac{Ax + B}{x^2 + 4} \, dx \)- \( \int \frac{Cx + D}{x^2 + 3} \, dx \)- \( \int \frac{Ex + F}{(x^2 + 3)^2} \, dx \)Using substitution methods and knowledge of integral tables will help evaluate these integrals.

Step 6: Combine Results

Sum the results from each separate integral to form the total integral solution. Ensure to include the constant of integration.

Key Concepts

Integration Techniques

Integration techniques are fundamental tools learned in calculus to solve integrals efficiently. Among these techniques, the method of partial fractions is particularly helpful when dealing with rational functions. In the problem at hand, we use partial fractions to decompose the complex rational function into simpler, easier-to-integrate parts. This technique involves expressing the integrand as a sum of fractions with simpler denominators. After decomposition, each term can be integrated separately using basic integration rules. Here's a quick rundown of the key steps involved:

• Identify and break down the integrand's denominator into simpler factors, assigning variables to unknown numerators.
• Combine and expand terms, matching coefficients to their respective powers of variables on both sides of the equation.
• Solve for the variables using algebraic or computational methods.

Remember, practicing with various rational functions will help you master this technique over time.

Rational Functions

Rational functions are expressions formed by dividing two polynomials. They are a central topic in calculus due to their versatility and occurrence in various mathematical problems. The given exercise involves a rational function where the numerator is a polynomial degree four and the denominator a product of quadratic polynomials. The decomposition into partial fractions helps simplify this complex expression. A good way to approach rational functions is to:

• Identify the degree and factors of both numerators and denominators.
• Learn specific decomposition rules, such as how to handle repeated factors in the denominator.
• Understand how different types of factors affect the resulting partial fractions (linear and quadratic).

Breaking them down enables us to analyze and integrate complex rational expressions effectively.

Calculus Problem Solving

Solving calculus problems often requires a combination of algebraic manipulation and conceptual understanding. In the case of our integral, the problem-solving process involves several deliberate steps, namely decomposition, expansion, coefficient matching, and then integration. Once the rational function is decomposed, it's crucial to solve the resulting system of linear equations correctly to find unknowns in the numerators. Here's how to solve such problems effectively:

• Approach the problem systematically—follow set steps without skipping any.
• Use algebraic techniques to simplify terms wherever possible.
• Employ computational tools when equations become too cumbersome to solve manually.
• Finally, double-check each integration step to ensure accuracy.

By incorporating these strategies, you can tackle calculus problems involving complex integrals with greater confidence and success.