Chapter 5-5: Mastering the Substitution Rule

Welcome to Chapter 5-5, where we'll be diving deep into the Substitution Rule, a powerful technique for evaluating integrals. This method, often called u-substitution, allows us to simplify complex integrals by changing the variable of integration.

Key Concepts:

  • The Idea: The substitution rule is essentially the chain rule in reverse. We aim to find a function $u = g(x)$ whose derivative, $g'(x)$, is also present (or can be easily obtained) in the integrand.
  • The Formula: If we have an integral of the form $\int f(g(x))g'(x) dx$, we can substitute $u = g(x)$ and $du = g'(x) dx$. The integral then becomes $\int f(u) du$, which is hopefully easier to evaluate.
  • Definite Integrals: When dealing with definite integrals, remember to change the limits of integration to be in terms of $u$ as well. If the original integral is $\int_a^b f(g(x))g'(x) dx$, then after the substitution $u = g(x)$, the new limits are $g(a)$ and $g(b)$, resulting in $\int_{g(a)}^{g(b)} f(u) du$.

Steps for U-Substitution:

  1. Choose u: Identify a suitable function $u = g(x)$ within the integrand. A good strategy is to look for a function whose derivative is also present. Consider functions inside other functions (like under a radical or in an exponent).
  2. Find du: Calculate the derivative of $u$ with respect to $x$, i.e., find $du = g'(x) dx$.
  3. Rewrite the Integral: Substitute $u$ and $du$ into the original integral. The goal is to get an integral entirely in terms of $u$.
  4. Evaluate the Integral: Evaluate the new integral with respect to $u$.
  5. Substitute Back: Replace $u$ with $g(x)$ to express the result in terms of the original variable $x$. Don't forget the constant of integration, +C, for indefinite integrals.
  6. Evaluate with New Limits (Definite Integrals): If you have a definite integral, remember to use the $u$ values for your new limits of integration.

Example:

Let's evaluate the integral $\int 2x \sqrt{1 + x^2} dx$.

  1. Choose u: Let $u = 1 + x^2$.
  2. Find du: Then $du = 2x dx$.
  3. Rewrite the Integral: The integral becomes $\int \sqrt{u} du = \int u^{1/2} du$.
  4. Evaluate the Integral: This is equal to $\frac{2}{3}u^{3/2} + C$.
  5. Substitute Back: Substituting back $u = 1 + x^2$, we get $\frac{2}{3}(1 + x^2)^{3/2} + C$.

Tips and Tricks:

  • Sometimes, you might need to manipulate the integrand slightly to get $du$ to match exactly. This may involve multiplying or dividing by a constant.
  • Practice is key! The more you practice, the better you'll become at recognizing suitable substitutions.
  • Don't be afraid to try different substitutions. If one doesn't work, try another.

Resources:

Here are the resources from class to help you further:

  • Section 5-5 PowerPoint (attached)
  • Video walkthrough (see below)

Keep practicing, and you'll master the substitution rule in no time! Good luck!