MATH 86857 Calculus of One Variable
Lecture 4 Notes
**Lecture 4: Techniques of Integration**
**Introduction:**
- Recap of previous lectures on the concept of integration and the fundamental theorem of
calculus.
- Emphasis on the importance of integration in calculus and real-world applications.
**Integration by Substitution:**
- Basic idea: Substituting a function inside another function to simplify the integration process.
- Steps to perform substitution: Choose an appropriate substitution, find the derivative, and
substitute back into the integral.
- Example problems demonstrating the technique of integration by substitution.
**Integration by Parts:**
- Formula: ∫u dv = uv - ∫v du
- Understanding the roles of u and dv in the integration by parts formula.
- Application of integration by parts to different types of functions.
- Examples illustrating the method of integration by parts.
**Trigonometric Integrals:**
- Review of basic trigonometric identities.
- Techniques for simplifying trigonometric integrals.
- Solving trigonometric integrals using substitution and integration by parts.
**Partial Fractions:**
- Decomposition of rational functions into partial fractions.
- Types of partial fraction decomposition: distinct linear factors, repeated linear factors, and
irreducible quadratic factors.
- Step-by-step process for finding the partial fraction decomposition of a rational function.
- Examples of integrating rational functions using partial fractions.
**Improper Integrals:**
- Definition of improper integrals and their properties.
- Types of improper integrals: infinite limits of integration and integrands with infinite
discontinuities.
- Evaluating improper integrals using limits.
- Examples of solving improper integrals.
**Conclusion:**
