AcademyIntegration
Academy
Substitution
Level 1 - Math I (Physics) topic page in Integration.
U-Substitution
U-substitution (also called reverse chain rule) is a technique for integrating composite functions. It reverses the chain rule from differentiation.
The Method
For \(\int f(g(x)) \cdot g'(x) \, dx\), let \(u = g(x)\), then \(du = g'(x) \, dx\):
U-Substitution
\[\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du\]
Steps
- Choose \(u\) as the inner function
- Compute \(du = u' \, dx\)
- Rewrite the integral in terms of \(u\)
- Integrate with respect to \(u\)
- Substitute back \(u = g(x)\)
Examples
Example 1: \(\int 2x \cos(x^2) \, dx\)
Let \(u = x^2\), then \(du = 2x \, dx\):
Example 1
\[\int 2x \cos(x^2) \, dx = \int \cos(u) \, du = \sin(u) + C = \sin(x^2) + C\]
Example 2: \(\int \frac{x}{\sqrt{x^2 + 1}} \, dx\)
Let \(u = x^2 + 1\), then \(du = 2x \, dx\):
Example 2
\[\int \frac{x}{\sqrt{x^2 + 1}} \, dx = \frac{1}{2} \int u^{-1/2} \, du = \frac{1}{2} \cdot 2u^{1/2} + C = \sqrt{x^2 + 1} + C\]
Definite Integrals with U-Substitution
When using u-substitution with definite integrals, you must change the limits:
Definite U-Sub
\[\int_a^b f(g(x)) g'(x) \, dx = \int_{u(a)}^{u(b)} f(u) \, du\]
Example: \(\int_0^1 2x(x^2 + 1)^3 \, dx\)
Definite Example
\[u = x^2 + 1, \quad du = 2x \, dx; \quad x = 0 \Rightarrow u = 1, \quad x = 1 \Rightarrow u = 2\]
Result
\[\int_1^2 u^3 \, du = \left[\frac{u^4}{4}\right]_1^2 = \frac{16}{4} - \frac{1}{4} = \frac{15}{4}\]