Problem 6 - Active Calculus

Question

For each of the following integrals involving rational functions, (1) use a CAS to find the partial fraction decomposition of the integrand; (2) evaluate the integral of the resulting function without the assistance of technology; (3) use a CAS to evaluate the original integral to test and compare your result in (2). a. \(\int \frac{x^{3}+x+1}{x^{4}-1} d x\) b. \(\int \frac{x^{5}+x^{2}+3}{x^{3}-6 x^{2}+11 x-6} d x\) c. \(\int \frac{x^{2}-x-1}{(x-3)^{3}} d x\)

Step-by-Step Solution

Answer

a. \[ \frac{1}{4}\ln|x-1| + \frac{1}{4}\ln|x| - \frac{1}{4}\ln|x+1| + \frac{1}{2}\ln|x^{2}+1| + C \] b. \[ \frac{x^{3}}{3} + \frac{7x^{2}}{2} - 47x + 84\ln|x-1| - 97\ln|x-2| + 60\ln|x-3| + C \] c. \[ \ln|x-3| - \frac{1}{x-3} + C \]

Step 1 Title - Use CAS for Partial Fraction Decomposition (Integral a)

Use a computer algebra system (CAS) to decompose \( \frac{x^{3}+x+1}{x^{4}-1} \) into partial fractions. The result is \[ \frac{x^{3}+x+1}{x^{4}-1} = \frac{1}{4(x-1)} + \frac{1}{4x} - \frac{1}{4(x+1)} + \frac{x}{(x^{2}+1)} \]

Step 2 Title - Evaluate the Integral (Integral a)

Integrate each partial fraction separately: \ \ \ 1. \[ \int \frac{1}{4(x-1)} dx = \frac{1}{4}\ln|x-1| + C \] \ 2. \[ \int \frac{1}{4x} dx = \frac{1}{4}\ln|x| + C \] \ 3. \[ \int \frac{-1}{4(x+1)} dx = -\frac{1}{4}\ln|x+1| + C \] \ 4. \[ \int \frac{x}{x^{2}+1} dx = \frac{1}{2}\ln|x^{2}+1| + C \] \ Combine all integrals to get the final result: \[ \frac{1}{4}\ln|x-1| + \frac{1}{4}\ln|x| - \frac{1}{4}\ln|x+1| + \frac{1}{2}\ln|x^{2}+1| + C \]

Step 3 Title - Use CAS for Validating Result (Integral a)

Use a CAS to directly evaluate the original integral \( \int \frac{x^{3}+x+1}{x^{4}-1} dx \). Verify that the CAS result matches the manually derived result: \[ \frac{1}{4}\ln|x-1| + \frac{1}{4}\ln|x| - \frac{1}{4}\ln|x+1| + \frac{1}{2}\ln|x^{2}+1| + C \]

Step 1 Title - Use CAS for Partial Fraction Decomposition (Integral b)

Use a CAS to decompose \( \frac{x^{5}+x^{2}+3}{x^{3}-6x^{2}+11x-6} \) into partial fractions. The result is: \[ \frac{x^{5}+x^{2}+3}{x^{3}-6x^{2}+11x-6} = x^{2} + 7x - 47 + \frac{84}{x-1} - \frac{97}{x-2} + \frac{60}{x-3} \]

Step 2 Title - Evaluate the Integral (Integral b)

Integrate each partial fraction separately: \ \ \ 1. \[ \int x^{2} dx = \frac{x^{3}}{3} + C \] \ 2. \[ \int 7x dx = \frac{7x^{2}}{2} + C \] \ 3. \[ \int -47 dx = -47x + C \] \ 4. \[ \int \frac{84}{x-1} dx = 84\ln|x-1| + C \] \ 5. \[ \int \frac{-97}{x-2} dx = -97\ln|x-2| + C \] \ 6. \[ \int \frac{60}{x-3} dx = 60\ln|x-3| + C \] \ Combine all integrals to get the final result: \[ \frac{x^{3}}{3} + \frac{7x^{2}}{2} - 47x + 84\ln|x-1| - 97\ln|x-2| + 60\ln|x-3| + C \]

Step 3 Title - Use CAS for Validating Result (Integral b)

Use a CAS to directly evaluate the original integral \( \int \frac{x^{5}+x^{2}+3}{x^{3}-6x^{2}+11x-6} dx \). Verify that the CAS result matches the manually derived result: \[ \frac{x^{3}}{3} + \frac{7x^{2}}{2} - 47x + 84\ln|x-1| - 97\ln|x-2| + 60\ln|x-3| + C \]

Step 1 Title - Use CAS for Partial Fraction Decomposition (Integral c)

Use a CAS to decompose \( \frac{x^{2}-x-1}{(x-3)^{3}} \) into partial fractions. The result is: \[ \frac{x^{2}-x-1}{(x-3)^{3}} = \frac{1}{(x-3)} + \frac{1}{(x-3)^{2}} \]

Step 2 Title - Evaluate the Integral (Integral c)

Integrate each partial fraction separately: \ \ \ 1. \[ \int \frac{1}{x-3} dx = \ln|x-3| + C \] \ 2. \[ \int \frac{1}{(x-3)^{2}} dx = -\frac{1}{x-3} + C \] \ Combine all integrals to get the final result: \[ \ln|x-3| - \frac{1}{x-3} + C \]

Step 3 Title - Use CAS for Validating Result (Integral c)

Use a CAS to directly evaluate the original integral \( \int \frac{x^{2}-x-1}{(x-3)^{3}}dx \). Verify that the CAS result matches the manually derived result: \[ \ln|x-3| - \frac{1}{x-3} + C \]

Key Concepts

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. An example is the integrand in the integral \(\frac{x^{3}+x+1}{x^{4}-1}\). These functions are important in calculus because they can often be simplified using partial fraction decomposition. By breaking down complex rational functions into simpler, more manageable components, we make integration much easier.

Integral Calculus

Integral calculus is the study of integrals and their properties. It allows us to find areas under curves, among other things. To handle integrals of rational functions, we often use partial fractions. For instance, the integral \(\frac{x^{3}+x+1}{x^{4}-1}\) can be decomposed into simpler fractions like \(\frac{1}{4(x-1)} \) and \(\frac{x}{(x^{2}+1)}\), which are easier to integrate.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a software tool that helps perform symbolic computations, including algebraic manipulations and calculus operations. CAS tools are immensely useful for finding the partial fraction decomposition of rational functions. For example, when dealing with the integral \(\frac{x^{3}+x+1}{x^{4}-1}\), a CAS can quickly decompose it. This makes the integration process easier and helps verify our results.

Integration Techniques

Various techniques can be used to integrate functions, especially rational ones. Partial fraction decomposition is a key technique here. Once we decompose a rational function into simpler fractions, we can integrate each part separately. For instance:\( \frac{1}{4(x-1)}\) integrates to \(\frac{1}{4}\text{ln}|x-1| + C\) and \(\frac{x}{x^{2}+1}\) integrates to \(\frac{1}{2}\text{ln}|x^{2}+1| + C\). Combining these results gives us the final integrated form.