Math 3B Exam 2 Review Notes

Definition: A method used to express rational functions as a sum of simpler fractions.
Purpose: Helps in simplifying the integration of rational functions.
Steps:
Factor the denominator completely.
Set up fractions with undetermined coefficients corresponding to each factor.
Multiply through by the common denominator to eliminate fractions.
Solve for the coefficients by equating coefficients on both sides.

Common Methods:
Direct Integration: For simple integral forms.
Integration by Parts: Useful when product of functions is involved.
Substitution: Change of variable simplifies the integral.
Trig Substitution: Often used for integrals involving square roots.
Example Techniques:
Recognize integrals that resemble standard forms (like exponential, trigonometric).
Apply limits of integration for definite integrals.

Definition: An integral is termed 'improper' if:
The interval of integration is infinite.
The integrand approaches infinity at any point within the interval.
Identifying Improper Integrals:
Examine limits when approaching infinite bounds or points of discontinuity.
Notation Example:
For integral [ \int_1^\infty \frac{1}{x^2} dx ], state it is improper due to infinite upper limit.
Evaluation Process:
Convert to a limit, e.g., [ \lim{b \to \infty} \int1^b \frac{1}{x^2} dx ].
Solve the limits to determine convergence/divergence.

Purpose: To determine the convergence/divergence of an integral by comparing it to a known integral.
Steps:
Identify a function that is easier to evaluate and has a similar form to the given integral.
Show that the original integral is less than or greater than the known integral.
Example Functions:
Compare ( rac{1}{x^2 + 1} ) with ( rac{1}{x^2} ) for large ( x ).
Ensure that conditions of the comparison test are satisfied.

Evaluate the following integrals:
( \int (6)(5) - (7) + 14 - 19 ) (check algebra & integration method)
( \int 4 cos(2x)sin(2x) dx ) (potential substitution)
( \int e^x \, dx ) (basic knowledge of exponential function)
( \int \sin(2x) tan(secx) dx ) (trigonometric integral)
Improper integral: ( \int_1^\infty \frac{8}{x^2 - 1} dx ) (investigate for convergence)

Practice these integration techniques and comparison tests with various functions to gain proficiency. Pay special attention to improper integrals and how to apply limits to evaluate them effectively.