Mathematics 136 Calculus 2: Partial Fractions and Improper Integrals - Exam 3 Solutions
Welcome to our comprehensive guide for solving problems related to partial fractions and improper integrals. This page provides detailed solutions for common exam questions in Calculus 2, specifically focusing on techniques for evaluating integrals using partial fraction decomposition and determining the convergence or divergence of improper integrals.
Partial Fractions
Partial fraction decomposition is a technique used to break down complex rational functions into simpler, more manageable terms that can be integrated individually. This method is particularly powerful when dealing with integrals of rational functions where the numerator and denominator are polynomials.
Steps for Partial Fraction Decomposition
- Ensure the numerator's degree is less than the denominator's degree (perform polynomial long division if necessary).
- Factor the denominator completely.
- Write down the appropriate partial fraction form based on the factors.
- Multiply through by the common denominator to clear fractions.
- Solve for the unknown coefficients.
- Integrate each term separately.
Types of Partial Fractions
The form of the partial fractions depends on the factors in the denominator:
- Linear factors (ax + b): Each factor yields a term in the form A/(ax + b)
- Repeated linear factors (ax + b): Results in terms A/(ax + b) + A/(ax + b) + ... + A/(ax + b)
- Irreducible quadratic factors (ax + bx + c): Produces terms of the form (Ax + B)/(ax + bx + c)
- Repeated quadratic factors (ax + bx + c): Leads to terms (Ax + B)/(ax + bx + c) + (Ax + B)/(ax + bx + c) + ... + (Ax + B)/(ax + bx + c)
Example 1: Simple Partial Fractions
Evaluate the integral:
Solution:
First, factor the denominator: x + 4x + 3 = (x + 1)(x + 3)
Set up the partial fractions:
Multiply through by (x + 1)(x + 3):
Expand and collect like terms:
By comparing coefficients:
- For x terms: A + B = 3
- For constant terms: 3A + B = 7
Solving the system of equations:
So the integral becomes:
Integrating each term:
Therefore, (3x + 7)/(x + 4x + 3) dx = 2 ln|x + 1| + ln|x + 3| + C
Example 2: Repeated Linear Factors
Evaluate the integral:
Solution:
First, factor the denominator: x + x - 2x = x(x + x - 2) = x(x + 2)(x - 1)
Set up the partial fractions:
Multiply through by x(x + 2)(x - 1):
Find the coefficients by substituting convenient values of x:
For x = 0: 3 = A(2)(-1) + 0 + 0 A = -3/2
For x = -2: 3 = 0 + B(-2)(-3) + 0 B = 1/2
For x = 1: 6 = 0 + 0 + C(1)(3) C = 2
So the integral becomes:
Integrating each term:
Therefore, (x + 2x + 3)/(x + x - 2x) dx = -3/2 ln|x| + 1/2 ln|x + 2| + 2 ln|x - 1| + C
Example 3: Quadratic Factors
Evaluate the integral:
Solution:
First, factor the denominator: x + x = x(x + 1)
Set up the partial fractions:
Multiply through by x(x + 1):
Comparing coefficients:
- For x terms: A + B = 1
- For x terms: C = 3
- For constant terms: A = 5
Solving: A = 5, C = 3, B = 1 - A = 1 - 5 = -4
So the integral becomes:
Integrating each term:
Therefore, (x + 3x + 5)/(x + x) dx = 5 ln|x| - 2 ln(x + 1) + 3 arctan(x) + C
Improper Integrals
Improper integrals are integrals where either the interval of integration is infinite or the integrand has a vertical asymptote within the interval of integration. To determine whether an improper integral converges or diverges, we often use limits.
Types of Improper Integrals
- Infinite interval: ^ f(x) dx = lim(b) f(x) dx or f(x) dx = lim(a-) f(x) dx
- Infinite interval (both ends): ^ f(x) dx = f(x) dx + ^ f(x) dx for some c in (-, )
- Discontinuous integrand: If f is discontinuous at a, f(x) dx = lim(ca) f(x) dx
- Discontinuous integrand: If f is discontinuous at b, f(x) dx = lim(cb) f(x) dx
Convergence Tests for Improper Integrals
| Test | Condition | Result |
|---|---|---|
| p-test (infinite interval) | ^ (1/x) dx | Converges if p > 1, diverges if p 1 |
| p-test (zero limit) | (1/x) dx | Converges if p < 1, diverges if p 1 |
| Comparison Test | If 0 f(x) g(x) and g(x) dx converges | Then f(x) dx converges |
| Limit Comparison Test | If lim(x) f(x)/g(x) = L where 0 < L < | Both integrals converge or both diverge |
Example 4: Infinite Interval Improper Integral
Determine whether the following improper integral converges or diverges, and evaluate it if it converges:
Solution:
Set up the limit:
Evaluate the integral:
Apply the limits:
Therefore, ^ (1/x) dx = 1 (converges)
Example 5: Discontinuous Integrand
Determine whether the following improper integral converges or diverges, and evaluate it if it converges:
Solution:
The integrand is undefined at x = 0. Set up the limit:
Evaluate the integral:
Apply the limits:
Therefore, (1/x) dx = 2 (converges)
Example 6: Infinite Interval with Comparison Test
Determine whether the following improper integral converges or diverges:
Solution using the Comparison Test:
For x 1, we have:
We know that ^ (2/x) dx = 2 ^ (1/x) dx converges (as p = 2 > 1 in the p-test).
Since our integrand is positive and less than the convergent integral 2/x, by the Comparison Test, the given integral also converges:
Example 7: Limit Comparison Test
Determine whether the following improper integral converges or diverges:
Solution using the Limit Comparison Test:
Let's compare with 1/x:
Since the limit is a positive finite number (1), both integrals either converge or diverge together.
We know that ^ (1/x) dx converges (by the p-test with p = 2 > 1).
Therefore, ^ (x + 1)/x dx also converges.
Tips for Solving Partial Fractions and Improper Integrals Problems
- Always factor the denominator completely when setting up partial fractions.
- Set up the correct form based on the types of factors in the denominator.
- When solving for unknown coefficients, substitute convenient values of x or compare coefficients.
- For improper integrals, identify the issue first (infinite interval or discontinuity).
- Always set up the limit properly when dealing with improper integrals.
- When using comparison tests, choose an appropriate function with known convergence properties.
- Remember the p-test results as they're frequently useful for comparison.
Common Mistakes to Avoid
- Failing to check if polynomial long division is needed before partial fractions.
- Setting up the wrong form for partial fractions (e.g., forgetting repeated factors).
- Forgetting to use limits when evaluating improper integrals.
- Misapplying the comparison test by not verifying the inequality conditions.
- Incorrectly assuming convergence without proper justification.
- Forgetting that ^ f(x) dx must be split into two improper integrals.
Practice Problems
Practice Problem 1
Evaluate the integral using partial fractions:
Hint: Start by factoring the denominator and setting up the appropriate partial fractions.
Practice Problem 2
Determine whether the following improper integral converges or diverges:
Hint: This is an improper integral of Type I (infinite interval). Set up the appropriate limit.
Practice Problem 3
Determine whether the following improper integral converges or diverges:
Hint: Consider using the Limit Comparison Test with an appropriate power function.
Practice Problem 4
Evaluate the integral using partial fractions:
Hint: This involves repeated factors. Set up your partial fractions accordingly.
Practice Problem 5
Determine whether the following improper integral converges or diverges:
Hint: This might involve integration by parts after setting up the limit.
Mathematics 136 Calculus 2 Partial Fractions And Improper Integrals Exam 3 Solutions and R...
Engineering Mathematics II Vector Calculus Fourier Analysis Partial Differential Equations...
Partial Differentiation And Multiple Integrals and Reference File Download Link
Using Definite Integrals Instead Of Indefinite Integrals In Solving Problems and Reference...