Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{x^{6}}{(x-2)^{2}(1-x)^{5}} d x\)

Short answer

Find coefficients through decomposition, integrate partial fractions individually, and sum results.

Step-by-step solution

Setup the Rational Function

To apply partial fraction decomposition, first express the integrand \( \frac{x^{6}}{(x-2)^{2}(1-x)^{5}} \) in terms of partial fractions. We assume a decomposition like:\[ \frac{x^{6}}{(x-2)^{2}(1-x)^{5}} = \frac{A}{x-2} + \frac{B}{(x-2)^{2}} + \frac{C}{(1-x)} + \frac{D}{(1-x)^{2}} + \frac{E}{(1-x)^{3}} + \frac{F}{(1-x)^{4}} + \frac{G}{(1-x)^{5}} \]

Express the Rational Function

Multiply both sides by \((x-2)^{2}(1-x)^{5}\) to eliminate the denominators. You get:\[x^6 = A(x-2)(1-x)^{5} + B(1-x)^{5} + C(x-2)^{2}(1-x)^{4} + D(x-2)^{2}(1-x)^{3} + E(x-2)^{2}(1-x)^{2} + F(x-2)^{2}(1-x) + G(x-2)^{2}\]

Simplify and Compare Coefficients

Expand and collect like terms (particularly powers of \(x\)) on both sides to equate coefficients. This will involve substituting specific values for \(x\) or solving a system of equations to find the coefficients \(A, B, C, D, E, F, G\).

Perform the Integration

Integrate each of the partial fractions separately using standard integration techniques. For each term like \( \frac{A}{x-2} \) and \( \frac{C}{(1-x)^{k}} \), integral formulas such as \( \int \frac{1}{x-a} dx = \ln|x-a| + C \) and \( \int (x-a)^n dx = \frac{(x-a)^{n+1}}{n+1} + C \) can be used. Address any negative powers of \((x-a)\) appropriately with logarithmic integration.

Combine the Results

Sum the integrals obtained from each of the partial fractions to get the final solution. Don't forget to include the constant of integration \(C\).

Key concepts

Integration Techniques

Integration is a fundamental concept in calculus, allowing us to find areas under curves or solve differential equations. When dealing with complex functions, not all integration tasks can be approached in the same way. For functions like rational functions, specific techniques such as partial fraction decomposition become essential.
Partial fraction decomposition is particularly useful for integrating rational functions, which are fractions where both the numerator and the denominator are polynomials. By breaking down the function into simpler parts—partial fractions—we can manage the integration more effectively. This approach simplifies the integral into a sum of simpler fractions, each easier to integrate.
The method involves expressing the complex rational function in terms of these partial fractions, solving for unknown coefficients, and then integrating each term separately. Only by employing these strategic integration techniques can we successfully solve otherwise cumbersome integrals.

Rational Functions

Rational functions are a vital class of functions in mathematics characterized by the ratio of two polynomials. These functions can take various forms, but the distinguishing feature is their polynomial numerator and denominator. Analytically, rational functions can illustrate numerous behaviors, including asymptotic tendencies and intercepts, which are crucial in calculus.
Integration of rational functions often requires special attention, especially when the degree of the numerator is equal to or greater than that of the denominator. In such cases, partial fraction decomposition can become indispensable. This involves separating the rational function into simpler, more manageable pieces.
These pieces, or partial fractions, allow us to integrate each section with relative ease compared to the original complex structure. Understanding the decomposition process is crucial for simplifying functions and further calculation, enhancing efficiency in analyzing rational functions.

Logarithmic Integration

Logarithmic integration is a specific technique within integration methods that comes into play when dealing with rational functions decomposed into terms involving the natural logarithm. It's employed primarily when a term in the integral is in the form of \(\frac{1}{x-a}\), which integrates to a logarithmic form.
This technique is quite straightforward: when you see a function that matches this pattern, you apply the integration rule \int \frac{1}{x-a} dx = \ln|x-a| + C\. During the decomposition of rational functions, some resulting partial fractions naturally fit this structure.
Effectively applying this rule allows for seamless integration of these terms, making what initially appears to be a complex integration task much simpler. Logarithmic integration simplifies by shifting the focus on identifying and managing terms that fit the logarithmic pattern, thus streamlining the integration process.