Substitution Rule

Concept Explanation

Substitution simplifies integration by introducing a new variable $u$ that maps a complex composite function into a basic integration form. For definite integrals, you must transform the integration boundaries ($a$ and $b$) to match the variable $u$ using the relation $u = g(x)$.

Visual / Geometric Intuition

Interactive Desmos Graph

Engineering Applications

Used to analyze frequency responses in signal processing, calculate electrical charging curves in RC circuits, and scale kinetic properties in thermodynamic simulations.

Example Problem

Evaluate the definite integral:

Solution

Let $u = x^2$. Then $du = 2x \, dx \implies x \, dx = \frac{1}{2} du$.

Step 1: Translate the integration boundaries

• When $x = 0 \implies u = 0^2 = 0$
• When $x = \sqrt{\ln 2} \implies u = (\sqrt{\ln 2})^2 = \ln 2$

Step 2: Substitute and integrate

$$\int_{0}^{\sqrt{\ln 2}} x e^{x^2} \, dx = \int_{0}^{\ln 2} e^u \left(\frac{1}{2} \, du\right)$$

$$= \frac{1}{2} \Big[ e^u \Big]_0^{\ln 2}$$

$$= \frac{1}{2} (e^{\ln 2} - e^0)$$

$$= \frac{1}{2} (2 - 1) = \frac{1}{2}$$

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Connections & References

• Parent Concepts: Definite Integrals
• Sub-concepts: Integration by Parts
• Course Links: Calc-2 Session 04