U-Substitution Review in Calculus 1

Review of U-Substitution in Calculus 1

General Overview

• U-Substitution is a technique revisited that comes from Calculus 1, specifically discussed in Section 5.5 of the textbook. It is employed at the beginning of Chapter 7, which covers anti-differentiation techniques including U-Substitution, Integration by Parts, etc.

• The lesson is delivered via video as it is deemed more effective than teaching about volumes of solids of revolution as found in Section 6.2.

• Hands-on materials: Handouts related to the lesson can be found on the D2L (Desire2Learn) News item and should be printed or reviewed prior to the lesson.

• Assignment Reminder: WebAssign related to this lesson will be due in the next class period, specifically on Tuesday during class time.

Fundamental Derivative Rules from Calculus 1

• A reminder of essential derivative formulas:

  • $rac{d}{dx}( ext{sin}(x)) = ext{cos}(x)$

  • $rac{d}{dx}( ext{cos}(x)) = - ext{sin}(x)$

• Each derivative rule has a corresponding inverse or anti-derivative rule:

  • The most general antiderivative (indefinite integral) of cosine is formulated as:

  • $ext{If } rac{d}{dx}( ext{sin}(x)) = ext{cos}(x), ext{ then } ext{anti-derivative is } ext{sin}(x) + C$

• To verify an antiderivative:

  • If you take the derivative of $ext{sin}(x) + C$, you should return to $ext{cos}(x)$, confirming the correctness.

Chain Rule Recap

• Composite Functions: Defined as a function created from two functions where one function is applied to the result of another. For example:

  • Let $f(x) = ext{sin}(x)$; if $g(x) = x^3$, then $f(g(x)) = ext{sin}(x^3)$.

• The derivative of a composite function follows the chain rule:

  • $rac{d}{dx}[f(g(x))] = f'(g(x)) imes g'(x)$

• Example of applying the chain rule:

  • For $rac{d}{dx}[ ext{cos}(x^3)]$:

  • Outside function is $f(u) = ext{cos}(u)$; derivative $f'(u) = - ext{sin}(u)$ evaluated at $u = g(x) = x^3$.

  • Inside function’s derivative $g'(x) = 3x^2$.

  • Therefore, $rac{d}{dx}[ ext{cos}(x^3)] = - ext{sin}(x^3) imes 3x^2$.

Revisiting U-Substitution

• Goal of U-Substitution: To simplify integrals of composite functions by changing the variable of integration. The substitution typically involves letting $u$ represent the inside function.

• Steps for U-Substitution:

  1. Select u: Let u be the inside function (e.g., if in $ext{sin}(2x)$, then let $u = 2x$).

  2. Differentiate u: Calculate the differential $du$;

  • For $u = 2x$, then $rac{du}{dx} = 2$ or $du = 2 imes dx$.

  1. Substitute Back: Replace all occurrences of the inside function in the integral with u and also replace $dx$ using the differential found in the previous step.

  2. Evaluate the Integral: Find the antiderivative in terms of u, then substitute back to the original variable.

Example of U-Substitution

• Consider the integral of $2 ext{cos}(2x)$.

  • Let $u = 2x$ then $du = 2dx$ or $dx = rac{du}{2}$.

  • Transform the integral:

  • $ext{Integral of } 2 ext{cos}(u) rac{du}{2} = ext{Integral of } ext{cos}(u) du$.

  • The antiderivative of $ext{cos}(u)$ is $ext{sin}(u) + C$.

  • Substitute back $u = 2x$:

  • Final result is $ext{sin}(2x) + C$.

Verification of U-Substitution Results

• To check your work, differentiate your answer:

  • If the derivative of $ext{sin}(2x) + C$ is taken, using the chain rule:

  • $rac{d}{dx}[ ext{sin}(2x) + C] = 2 ext{cos}(2x)$,

  • Which confirms the correctness of the integral as it matches the original function's derivative.

Conclusion

• U-Substitution is essential for effectively integrating composite functions. Proper execution of this technique shapes the foundation for advanced integration strategies later in calculus studies. Familiarity with derivative rules and the chain rule enhances understanding and application of U-Substitution.ok