Okay, I need to create a professional and useful wiki document about partial fraction decomposition in Korean. The user provided a structure and some requirements, so I should follow that closely. Let me by understanding the topic.

fraction decomposition is a method used in calculus to break down complex rational functions into simpler fractions, which makes integration easier. The user specified the categories as Education > Mathematics > Calculus, so the document should be technical but accessible to students studying calculus.

First, the title should be in H1. The main title is "분수분해" which is the Korean term for partial fraction decomposition. The introduction should give a general overview, explaining what it is and its purpose. I should mention that it's used to integrate rational functions by breaking them into simpler parts.

Next, the detailed sections. The structure should have H2 and H3 headings. Let me think about the main sections. Maybe start with the definition and purpose, then the process steps, examples, applications, and related concepts. Each of these can be an H2 section.

Under the process steps, breaking it down into H3 headings like factorizing the denominator, setting up the decomposition form, solving coefficients, and integrating. Each step needs a clear explanation. For example, factorizing the denominator into linear and irreducible quadratic factors. Then, for each type of factor, how the decomposition looks. Then, solving for A, B, C using equations. Maybe include an example here.

Examples are important. I should include a simple example and a more complex one. Let's take (x^2 + 3x + 2)/(x^3 - x) as an example. Factor the denominator into x(x-1)(x+1), set up the partial fractions, solve for coefficients. Show the steps algebraically.

Applications section should mention integration, inverse Laplace transforms, solving differential equations. This shows the practical use of partial fractions.

Related concepts might include polynomial long division, rational functions, irreducible polynomials. These are prerequisites or related topics.

References or related documents section can list textbooks or other wiki entries. Maybe include a table here with some standard references.

I need to ensure that the content is accurate. Let me recall the steps for partial fraction decomposition. First, if the degree of the numerator is higher than the denominator, perform polynomial division. Then factor the denominator. Then express as a sum of partial fractions. Solve for coefficients using substitution or equating coefficients.

I should mention cases with repeated factors. For example, if the denominator has (x-a)^2, then the decomposition includes A/(x-a) + B/(x-a)^2. Also, irreducible quadratic factors would have (Ax + B)/(quadratic).

In the examples, maybe start with a simple case without repeated or quadratic factors. Then a second example with a repeated linear factor. For instance, (x+1)/(x^2(x-1)).

Use markdown formatting. H1 for title, H2 for sections like 개요, 분수분해의 정의와 목적, 분해 과정, 활용 분야, 관련 개념. H3 for sub-sections under 분해 과정 like 분모 인수분해, 분해 형태 설정, 계수 결정, 적분 적용.

In the examples, use code blocks for mathematical expressions with LaTeX syntax. For instance:

$$ \frac{x^2 + 3x + 2}{x(x-1)(x+1)} = \frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1} $$

Then show the steps to find A, B, C. Multiply both sides by denominator to get:

x^2 + 3x + 2 = A(x-1)(x+1) + Bx(x+1) + Cx(x-1)

Then substitute x=0, x=1, x=-1 to find A, B, C.

For the example with repeated factors:

$$ \frac{x+1}{x^2(x-1)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x-1} $$

Then solve similarly.

Applications: When integrating ∫(x^2 + 3x + 2)/(x^3 - x) dx, after decomposition, integrate each term separately.

Also mention inverse Laplace transforms where partial fractions are used to break down the transform into simpler terms that can be inverted using tables.

Make sure to explain any technical terms. For example, define rational function as a ratio of two polynomials, irreducible quadratic factors as those that can't be factored into real linear terms.

Check the length. The document should be at least 800 characters. Each section adds to that. The examples and steps should provide enough detail.

Avoid markdown errors. Use proper syntax for headers, lists, and equations. Ensure that LaTeX is properly enclosed in $$ for display equations.

Now, putting it all together. Start with the title, then the overview, then each section as outlined. Use examples to illustrate the process. Mention applications and related concepts. Add references if possible, but the user might not expect actual citations, just a section for them.

Wait, the user said to include a "참고 자료" or "관련 문서" section. Maybe list some calculus textbooks or online resources. Since it's a wiki, perhaps link to related entries like 미적분학, 유리 함수, 다항식 나눗셈, 라플라스 역변환.

Also, ensure that the language is Korean, using the correct terminology. For example, "partial fraction decomposition" is "분수분해" in Korean. Check if there's a more precise term, but according to the user's input, it's 분수분해.

Double-check the steps in the examples. For the first example, after substituting x=0, x=1, x=-1, calculate the coefficients correctly. For instance, when x=0, left side is 0^2 +3*0 +2=2. Right side is A(-1)(1) + 0 +0 => A(-1) = 2 => A=-2. Wait, that doesn't seem right. Let me recalculate.

Wait, original example: numerator x² +3x +2 over denominator x(x-1)(x+1). So the equation is x² +3x +2 = A(x-1)(x+1) + Bx(x+1) + Cx(x-1).

At x=0: 0 +0 +2 = A(-1)(1) + 0 +0 => 2 = -A => A = -2.

At x=1: 1 +3 +2 = A(0)(2) + B(1)(2) + C(1)(0) => 6 = 2B => B=3.

At x=-1: 1 -3 +2 = A(-2)(0) + B(-1)(0) + C(-1)(-2) => 0 = 2C => C=0.

Wait, then the decomposition would be -2/x +3/(x-1) +0/(x+1). But when combining, let's check:

-2/x +3/(x-1) = (-2(x-1) +3x)/x(x-1) = (-2x +2 +3x)/x(x-1) = (x+2)/x(x-1). Multiply by (x+1) denominator? Wait, the original denominator is x(x-1)(x+1), so the combined numerator should be -2(x-1)(x+1) +3x(x+1) +0.

Let me compute that:

-2(x² -1) +3x(x+1) = -2x² +2 +3x² +3x = x² +3x +2, which matches the original numerator. So the decomposition is correct. So the example works. Then integrating term by term would be -2 ln|x| +3 ln|x-1| + C, etc.

So the example is valid. Just make sure the steps are correctly shown.

Another example with repeated factors. Let's take (x+1)/(x^2(x-1)). The decomposition is A/x + B/x² + C/(x-1). Multiply both sides by x²(x-1):

x+1 = A x(x-1) + B(x-1) + C x².

To find A, B, C, substitute x=0: 0+1 = 0 + B(-1) +0 => 1 = -B => B=-1.

x=1: 1+1 = A(1)(0) + B(0) + C(1) => 2=C.

Then expand the equation:

x+1 = A x² -A x + Bx - B + C x².

Combine like terms:

x+1 = (A + C)x² + (-A + B)x - B.

Since B=-1, then coefficients:

A + C = 0 (since there's no x² term on left)

-A + B = 1 (coefficient of x)

-B =1 (constant term). Wait, -B = -(-1)=1, which matches the constant term on the left (1). So the coefficients:

A + C=0

-A + (-1) =1 → -A = 2 → A=-2

Then C= -A=2.

So decomposition is (-2)/x + (-1)/x² +2/(x-1). Which can be integrated accordingly.

These examples should help illustrate the process.

I think that covers the main points. Now structure the document accordingly in Korean, making sure each section flows logically and provides clear explanations with examples and steps. Use markdown syntax properly, including headers, bullet points, code blocks for equations, and any necessary tables or emphasis.