Overview
This lecture reviews the u-substitution method for integration, illustrating it as the inverse process of the chain rule and demonstrating step-by-step solutions to various integral problems using this technique.
U-Substitution Basics
- U-substitution reverses the chain rule for finding antiderivatives.
- Identify an "inside function" (u) within a composite function and set u equal to that expression.
- Compute du by differentiating u with respect to x (du = derivative × dx).
- Rewrite the integral entirely in terms of u and du before integrating.
Simple U-Substitution Examples
- For integrals like ∫2x cos(x²) dx, set u = x², du = 2x dx, yielding ∫cos(u) du = sin(x²) + C.
- It is crucial to substitute so no x remains after changing variables.
Adjusting du with Constants
- If du does not directly match other terms, adjust by multiplying/dividing both sides by a constant.
- Example: For ∫x²sec²(x³) dx, set u = x³, du = 3x² dx, rewrite x² dx = (1/3)du, integrate as (1/3)tan(x³) + C.
Power and Rational Integrals
- For ∫(x+2)/(x²+4x+3)³ dx, set u = x²+4x+3, du = (2x+4)dx, adjust to x+2 dx = (1/2)du, then integrate using the power rule.
Definite Integrals and Changing Limits
- For definite integrals, you can:
- Substitute back to x after integrating in terms of u, or
- Change the limits to be in terms of u by plugging in the original x-limits into u.
- Both methods give the same result, e.g., ∫₀^{π/2} e^{sin x} cos x dx → e-1.
Less Obvious U Choices and Algebraic Manipulations
- If an extra x remains, express it in terms of u, such as using x = u-1 if u = x+1.
- More complex forms may require expanding and combining like terms after substitution.
More Examples and Techniques
- Integrals involving roots or powers, e.g., ∫x√(x+1) dx, require expressing all x in terms of u after substitution.
- For ∫sec²(√x)/√x dx, set u = √x, du = (1/2)x^{-1/2} dx, manipulate to form the integral in du.
Key Terms & Definitions
- U-substitution — A technique to simplify integration by substituting u for an inner function.
- Chain rule — Differentiation method for composite functions: derivative of outer × derivative of inner.
- Antiderivative — The reverse of differentiation; a function whose derivative is the given function.
- Definite integral — Integral with upper and lower limits; produces a numerical result.
- Indefinite integral — Integral without bounds; includes an arbitrary constant (C).
Action Items / Next Steps
- Practice various u-substitution problems.
- Review chain rule and power rule for integration.
- Complete assigned homework on u-substitution integrals.