Partial Fraction Decomposition: A Detailed Example

Hey guys! Today, we're diving deep into the fascinating world of indefinite integrals, specifically focusing on how to tackle integrals that involve rational functions. We'll be dissecting a problem that might seem daunting at first glance, but with the right techniques, we'll break it down into manageable pieces. Our mission? To evaluate the indefinite integral:

$$\int \frac{3x^3 + 7x^2 + 8x + 12}{x^4 + 4x^2} dx$$

This looks like a beast, right? But don't worry, we've got a secret weapon: partial fraction decomposition. This technique allows us to rewrite complex rational functions into simpler fractions that are much easier to integrate. So, let's roll up our sleeves and get started!

The Power of Partial Fraction Decomposition

The core idea behind partial fraction decomposition is to express a rational function (a fraction where the numerator and denominator are polynomials) as a sum of simpler fractions. This is incredibly useful because integrating simpler fractions is often much easier than integrating the original complex one. In our case, the integrand is:

$$\frac{3x^3 + 7x^2 + 8x + 12}{x^4 + 4x^2}$$

The problem statement tells us that this fraction can be decomposed into the following form:

$$\frac{a}{x^2} + \frac{b}{x} + \frac{cx + d}{x^2 + 4}$$

where a, b, c, and d are constants that we need to determine. This is where the fun begins! Our goal is to find these mystery constants. To do this, we'll follow a series of algebraic manipulations.

Finding the Common Denominator

The first step is to combine the fractions on the right-hand side of the equation into a single fraction. To do this, we need to find a common denominator, which in this case is $x^2(x2 + 4)$. We multiply each fraction by the appropriate factors to achieve this common denominator:

$$\frac{a}{x^2} \cdot \frac{x^2 + 4}{x^2 + 4} + \frac{b}{x} \cdot \frac{x(x^2 + 4)}{x(x^2 + 4)} + \frac{cx + d}{x^2 + 4} \cdot \frac{x^2}{x^2}$$

This gives us:

$$\frac{a(x^2 + 4)}{x^2(x^2 + 4)} + \frac{bx(x^2 + 4)}{x^2(x^2 + 4)} + \frac{(cx + d)x^2}{x^2(x^2 + 4)}$$

Now we can combine the numerators over the common denominator:

$$\frac{a(x^2 + 4) + bx(x^2 + 4) + (cx + d)x^2}{x^2(x^2 + 4)}$$

Equating Numerators: The Key Step

Since the denominators on both sides of our equation are now the same, we can equate the numerators. This is a crucial step because it transforms our fractional equation into a polynomial equation, which is much easier to work with. So, we have:

$$3x^3 + 7x^2 + 8x + 12 = a(x^2 + 4) + bx(x^2 + 4) + (cx + d)x^2$$

Now, let's expand the right-hand side of the equation:

$$3x^3 + 7x^2 + 8x + 12 = ax^2 + 4a + bx^3 + 4bx + cx^3 + dx^2$$

Next, we'll group the terms with the same powers of x:

$$3x^3 + 7x^2 + 8x + 12 = (b + c)x^3 + (a + d)x^2 + 4bx + 4a$$

This is where the magic happens! For these two polynomials to be equal, the coefficients of the corresponding powers of x must be equal. This gives us a system of linear equations that we can solve for a, b, c, and d.

Solving for the Constants: A System of Equations

By equating the coefficients, we get the following system of equations:

1. Coefficient of $x^3$: $b + c = 3$
2. Coefficient of $x^2$: $a + d = 7$
3. Coefficient of $x$: $4b = 8$
4. Constant term: $4a = 12$

This system looks much less intimidating than the original integral, doesn't it? We can solve this system using various methods, such as substitution or elimination. Let's start with the easiest equations.

From equation (4), we can directly solve for a:

$$4a = 12 \implies a = \frac{12}{4} = 3$$

So, we've found our first constant! Now, let's move on to equation (3) to solve for b:

$$4b = 8 \implies b = \frac{8}{4} = 2$$

Great! We've got a and b. Now, we can use these values to find c and d. From equation (1), we have:

$$b + c = 3 \implies 2 + c = 3 \implies c = 3 - 2 = 1$$

And finally, from equation (2), we can solve for d:

$$a + d = 7 \implies 3 + d = 7 \implies d = 7 - 3 = 4$$

We did it! We've found all the constants: a = 3, b = 2, c = 1, and d = 4. Now, let's plug these values back into our partial fraction decomposition.

The Decomposed Integrand: A Simpler Form

Substituting the values of a, b, c, and d into our partial fraction decomposition, we get:

$$\frac{3}{x^2} + \frac{2}{x} + \frac{x + 4}{x^2 + 4}$$

This looks much more manageable than the original integrand, doesn't it? Now, we can rewrite our original integral as a sum of simpler integrals:

$$\int \frac{3x^3 + 7x^2 + 8x + 12}{x^4 + 4x^2} dx = \int \left( \frac{3}{x^2} + \frac{2}{x} + \frac{x + 4}{x^2 + 4} \right) dx$$

We can now split this integral into three separate integrals:

$$\int \frac{3}{x^2} dx + \int \frac{2}{x} dx + \int \frac{x + 4}{x^2 + 4} dx$$

Tackling the Simpler Integrals

Let's tackle these integrals one by one.

1. $$\int \frac{3}{x^2} dx = 3 \int x^{-2} dx = 3 \cdot \frac{x^{-1}}{-1} + C_1 = -\frac{3}{x} + C_1$$

2. $$\int \frac{2}{x} dx = 2 \int \frac{1}{x} dx = 2 \ln|x| + C_2$$

3. The third integral, $\int \frac{x + 4}{x^2 + 4} dx$, requires a little more finesse. We can split it into two integrals:

   $$\int \frac{x}{x^2 + 4} dx + \int \frac{4}{x^2 + 4} dx$$

   For the first part, $\int \frac{x}{x^2 + 4} dx$, we can use a u-substitution. Let $u = x^2 + 4$, then $du = 2x dx$, so $\frac{1}{2} du = x dx$. Thus,

   $$\int \frac{x}{x^2 + 4} dx = \frac{1}{2} \int \frac{1}{u} du = \frac{1}{2} \ln|u| + C_3 = \frac{1}{2} \ln(x^2 + 4) + C_3$$

   For the second part, $\int \frac{4}{x^2 + 4} dx$, we can use the arctangent integral formula. Recall that $\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$. In our case, $a^2 = 4$, so $a = 2$. Therefore,

   $$\int \frac{4}{x^2 + 4} dx = 4 \int \frac{1}{x^2 + 4} dx = 4 \cdot \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C_4 = 2 \arctan\left(\frac{x}{2}\right) + C_4$$

   Combining these two parts, we get:

   $$\int \frac{x + 4}{x^2 + 4} dx = \frac{1}{2} \ln(x^2 + 4) + 2 \arctan\left(\frac{x}{2}\right) + C_3 + C_4$$

The Final Solution: Putting It All Together

Now, let's combine the results of all three integrals:

$$\int \frac{3x^3 + 7x^2 + 8x + 12}{x^4 + 4x^2} dx = -\frac{3}{x} + 2 \ln|x| + \frac{1}{2} \ln(x^2 + 4) + 2 \arctan\left(\frac{x}{2}\right) + C$$

where $C = C_1 + C_2 + C_3 + C_4$ is the constant of integration. And there you have it! We've successfully evaluated the indefinite integral using partial fraction decomposition.

Key Takeaways: Mastering Partial Fractions

This journey through partial fraction decomposition highlights some crucial concepts:

• Partial fraction decomposition is a powerful technique for simplifying rational functions, making them easier to integrate.
• The process involves breaking down a complex fraction into a sum of simpler fractions with denominators that are factors of the original denominator.
• Solving for the unknown constants requires equating numerators and solving a system of linear equations.
• Remember to handle repeated factors and irreducible quadratic factors appropriately in the decomposition.
• Don't forget the constant of integration! It's a crucial part of any indefinite integral.

So, the next time you encounter a tricky integral involving a rational function, remember the power of partial fraction decomposition. With a little practice, you'll be able to conquer even the most challenging integrals. Keep practicing, and you'll become a master of integration in no time!

Now, to specifically answer the original question about the values of a and b:

$$a = 3$$

$$b = 2$$

We found these values while solving the system of equations derived from the partial fraction decomposition. These constants are essential for rewriting the original integrand into a form that we can easily integrate. Understanding how to find these constants is the key to mastering this technique. So, keep practicing and you'll be a pro in no time!