Simplifying Integrals
U-Substitution is the tool we use to integrate nested functions. It undoes the Chain Rule. We pick a part of the function to be \(u\), find \(du\), and rewrite the entire integral in terms of \(u\).
Worked Examples
Example 1: Basic Substitution
Evaluate \(\int 2x \cos(x^2) dx\).
- Let \(u = x^2\). Then \(du = 2x dx\).
- The integral becomes \(\int \cos(u) du\).
- Integrate: \(\sin(u) + C\).
- Substitute back: \(\sin(x^2) + C\).
- Result: \(\sin(x^2) + C\).
Example 2: With Limits
Evaluate \(\int_0^1 (2x+1)^3 dx\).
- Let \(u = 2x+1\). Then \(du = 2dx \implies dx = \frac{du}{2}\).
- Change limits: if \(x=0, u=1\). If \(x=1, u=3\).
- New integral: \(\int_1^3 u^3 \frac{du}{2} = \frac{1}{2} \int_1^3 u^3 du\).
- Integrate: \(\frac{1}{2} [\frac{u^4}{4}]_1^3 = \frac{1}{8} (81 - 1) = \frac{80}{8} = 10\).
- Result: 10.
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often use substitution to solve for Wavepackets. A wavepacket is a sum of many different frequencies. By substituting coordinates (like moving from \(x\) to a "shifted" coordinate \(x - vt\)), we can see how the packet moves through space. This "change of variables" is the mathematical equivalent of moving into a frame of reference that travels along with the particle.