Overview
This lecture introduces the substitution method (also called u-substitution) as a key technique for solving integrals that cannot be computed directly.
Review of Integral Basics
- The basic power rule for integrals: ∫xⁿ dx = (1/(n+1)) xⁿ⁺¹ + C, where n ≠ -1.
- Direct integration isn't possible for certain complex products or composite functions.
Substitution (u-Substitution) Method
- The substitution method rewrites an integral by changing the variable (from x to u) to simplify the expression.
- Choose u so that its derivative (du/dx) appears elsewhere in the integral.
- Replace the original variable and differential with u and du in the integrand.
Worked Examples
- For ∫2x(x²+5)⁵ dx, let u = x²+5, so du = 2x dx; the integral becomes ∫u⁵ du.
- Integrate using the formula: ∫u⁵ du = (1/6)u⁶ + C, then substitute back for x.
- For ∫[6x²+6]/√(x³+3x+1) dx, let u = x³+3x+1, so du = (3x²+3)dx; adjust with constants to match numerator.
- Express everything in terms of u and du, integrate, and substitute back to x.
Definite Integrals and Changing Limits
- When using substitution in a definite integral, convert limits from x to u by plugging values into the u equation.
- Integrate the new function with respect to u, then evaluate at the new limits.
Tips for Applying Substitution
- Select u so that its derivative matches, or is a multiple of, the remaining parts of the integrand.
- Adjust constants as needed to match du terms.
- Substitute back for x at the end for indefinite integrals; change limits for definite integrals.
Key Terms & Definitions
- Integral — The area under the curve of a function, found by anti-differentiation.
- u-Substitution — A method where part of the integrand is replaced with u to simplify integration.
- Definite Integral — An integral with upper and lower bounds, giving a numerical result.
- Indefinite Integral — An integral without bounds, resulting in a general formula plus C.
Action Items / Next Steps
- Practice more problems using the substitution method.
- Review basic integral rules and ensure understanding of differentiation.
- Anticipate discussion of more complex substitution examples in future sessions.