Calc 1: Section 5.5 - U-Substitution

Welcome to Section 5.5, where we'll explore the powerful technique of U-Substitution! U-Substitution is essentially the reverse of the chain rule, allowing us to tackle more complex integrals. It's all about recognizing a function and its derivative (or a multiple thereof) within the integral.

The Substitution Rule

The core idea behind U-Substitution, also known as the substitution rule, can be stated as follows:

If $u = g(x)$ is a differentiable function whose range is an interval I, and $f$ is continuous on I, then:

$$\int f(g(x))g'(x) dx = \int f(u) du$$

How to Apply U-Substitution

  1. Choose your 'u': Look for a part of the integrand whose derivative (or a constant multiple of it) is also present in the integrand. This is often the 'inner' function of a composite function.
  2. Find du: Calculate the derivative of your chosen 'u' with respect to x (i.e., $du/dx$) and solve for $du$.
  3. Substitute: Replace the original expression in the integral with 'u' and 'du'. Your integral should now be entirely in terms of 'u'.
  4. Integrate: Evaluate the new integral with respect to 'u'.
  5. Back-Substitute: Replace 'u' with its original expression in terms of 'x'. Don't forget the constant of integration, '+ C' !

Examples

Let's walk through a few examples to illustrate U-Substitution in action:

Example 1:

Evaluate $\int x^3 \cos(x^4 + 2) dx$

Let $u = x^4 + 2$. Then, $du = 4x^3 dx$. Solving for $dx$, we get $dx = \frac{du}{4x^3}$.

Substituting, we have: $\int x^3 \cos(u) \frac{du}{4x^3} = \frac{1}{4} \int \cos(u) du$

Integrating, we get: $\frac{1}{4} \sin(u) + C$

Back-substituting, we have: $\frac{1}{4} \sin(x^4 + 2) + C$

Example 2:

Evaluate $\int \sqrt{2x + 1} dx$

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

Substituting, we get: $\int \sqrt{u} \cdot \frac{1}{2} du = \frac{1}{2} \int u^{1/2} du$

Integrating, we get: $\frac{1}{2} \cdot \frac{u^{3/2}}{3/2} + C = \frac{1}{3} u^{3/2} + C$

Back-substituting, we have: $\frac{1}{3} (2x + 1)^{3/2} + C$

Example 3:

Evaluate $\int e^{5x} dx$

Let $u = 5x$. Then, $du = 5 dx$, so $dx = \frac{1}{5} du$.

Substituting, we get: $\int e^u \cdot \frac{1}{5} du = \frac{1}{5} \int e^u du$

Integrating, we get: $\frac{1}{5} e^u + C$

Back-substituting, we have: $\frac{1}{5} e^{5x} + C$

Practice Makes Perfect!

U-Substitution might seem tricky at first, but with practice, you'll become more comfortable identifying suitable 'u' values and simplifying integrals. Keep practicing, and you'll master this essential integration technique in no time!