Unit 5 Lesson 2 - U Substitution-Long Division-Completing the Square

Topic 6.9: Integrating Using Substitution

  • Key Concept: U-substitution is a method used in integration, essential for helping simplify the process of finding integrals.

    • Objective: Look for a part of the function whose derivative is also present in the function (can be off by a coefficient).

    • Comparison: U-substitution plays a similar role in integration as the Chain Rule does in differentiation.

Procedure for Using U-Substitution

  1. Identify Component:

    • Look for a segment in the function whose derivative exists elsewhere in the function.

    • Common choices include the denominator or parts raised to a power.

  2. Define u:

    • Set ( u = g(x) ) and then differentiate (find ( du )).

  3. Substitute:

    • Replace the identified parts of the integral with ( u ) and ( du ) to make the integral easier to solve.

Substitution Method for Indefinite Integrals

  • Objective: U-substitution aims to "undo" the Chain Rule in differentiation.

    1. Let ( u = g(x) )

    2. Find ( du = g'(x)dx )

    3. Find the antiderivative of f

    4. Substitute back to return to the variable x.

General Power Rule for Integration

  • If ( g ) is differentiable, then:[ \int g(x)^n g'(x) , dx = \frac{g(x)^{n+1}}{n+1} + C, \quad n
    eq -1 ]

    • Alternatively:[ \int u^n , du = \frac{u^{n+1}}{n+1} + C, \quad n
      eq -1 ]

Problem Set Examples (Page 2)

  • Integral Problems for practice:A. ( \int \frac{x^2 + 1}{2x} , dx )B. ( \int 4 \cos(4x) , dx ) C. ( \int \frac{8x}{4x^2 + 1} , dx ) D. ( \int \frac{x}{x^2 + 2} \frac{3}{, dx} ) E. ( \int \frac{x}{4x^2 + 3} , dx )F. ( \int \frac{3}{7 - 2x^3} , \frac{4x^2}{dx} )

Trigonometric Identities and Integration (Page 3)

  • Example Problems: Use trigonometric identities and algebra to prepare functions for u-substitution before integrating.

    • Example integrals involve:

      • ( an(x) , dx )

      • ( ext{cot}(x) , dx )

      • ( ext{sec}(x) , dx )

      • ( ext{csc}(x) , dx )

Changing Limits of Integration (Page 4)

  • When using u-substitution on definite integrals, change the limits to match the transformed function:

    • From ([a, b] ) in x to ([g(a), g(b)] ) in u.

    • Integral representation:[ \int_a^b f(g(x)) g'(x) , dx = \int_{g(a)}^{g(b)} f(u) , du ]

Example: Definite Integrals with Limits

  • A. ( \int_0^{\frac{\pi}{4}} \tan(x) \sec^2(x) , dx )

  • B. ( \int_0^{\frac{\pi}{6}} \sin(2\theta) \sec^3(\theta) , d\theta )

  • C. ( \int_0^2 \frac{3x^2}{1 + x^3} , dx )

  • D. ( \int_0^1 \frac{x}{1 + 5x^2} , dx )

Example Problems with Long Division (Page 5)

  • Using Long Division: Useful for integration when the degree of the numerator is greater than the degree of the denominator.

  • Completing the Square: Helps in rewriting integrands for easier integration, particularly when dealing with second-degree polynomials.

Problem Set 7: Working the Chain Rule Backwards (Page 6)

  • To undo the Chain Rule: The antiderivative must adjust with a constant factor.

  • Problems involve rewriting expressions to integrate effectively using identities.

  • Examples of integrals such as ( an(x) ) can be re-expressed in terms of sine and cosine for easier calculation.

Problem Set 8: The Power of One (Page 7)

  • Focus on minor adjustments needed for certain integrals.

  • Examples of how to evaluate functions with specific integrands and using adjustments effectively.

  • Practice includes evaluating integrals with adjustments like ( x^{-1} ) or polynomials transformed into manageable forms.