Notes on the Fundamental Theorem of Calculus and Integration Techniques

Fundamental Theorem of Calculus (FTC)

  • Understanding FTC: Comprises two parts that relate differentiation and integration.

Part 1 of the FTC

  • If $g(x) = \int_{a}^{x} f(t) \, dt$, then

    • $g'(x) = f(x)$ where $f$ is continuous on an interval.

Part 2 of the FTC

  • If $F$ is an antiderivative of $f$, then:

    • $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$.

Worksheet 1: Fundamental Theorem

  1. Integration and Derivatives:
    a. Find $g'(x)$ for $g(x) = \int_{a}^{x} 2 + \cos(t) \, dt$

    • Apply Part 1 and differentiate:

      • $g'(x) = 2 + \cos(x)$.

    b. Evaluate the integral using Part 2, then differentiate:

    • Evaluate $g'(x)$ from $g(x)$ then derive if needed.

  2. Derivative Examples:
    a. $y = \int_{17}^{x} \tan(t) \sin(t^4) \, dt$

    • Apply Part 1:

      • $y' = \tan(x) \sin(x^4)$.

    b. $y = \int_{5}^{x} \frac{1}{u^2 - 5} \, du$

    • Apply Part 1:

      • $y' = \frac{1}{x^2 - 5}$.

    c. $g(t) = \int_{t}^{t^2} \frac{s^2}{s^2 + 1} \, ds$

    • Use the Leibniz Rule:

      • $g'(t) = \frac{(t^2)^2}{(t^2)^2 + 1} \cdot 2t - \frac{t^2}{t^2 + 1}$.

  3. Using Part 2 to Evaluate Integrals:
    a. $\int_{2}^{4} (3x + 5) \, dx$

    • Evaluate: $\left[\frac{3x^2}{2} + 5x\right]_{2}^{4}$.
      b. Verify the divergent nature of integrals when applicable.

Worksheet 2: Indefinite Integrals

  1. Finding Integrals:
    a. $\int x^2 \cdot e^{x} \, dx$

    • Apply integration by parts.
      b. $\int\frac{1}{t^2} - \frac{1}{t^4} \, dt$

    • Result: $-\frac{1}{t} + \frac{1}{3t^3}$ + C.

  2. Substitution Technique:
    a. Solve and demonstrate for $\int x^x \, dx$ using substitution.
    b. More portfolio integrals involving trigonometric functions, algebraic expressions.

Worksheet 3: Integration Techniques

  1. Evaluate Integrals:
    a. $\int x \sec(x) \tan(x) \, dx$

    • Resulting form after integrating by recognizing derivatives and patterns.
      b. Use trigonometric substitutions in integrals involving trig functions and power expansions.

  2. Integration by Parts Examples:
    a. General practice problems should illustrate key theory like LIATE and simplifying integrals using known derivatives.

Worksheet 4: Partial Fractions

  1. Decomposing Fractions:

    • Use general template for polynomial and rational functions.

    • Show solutions for integrals like $\int \frac{x^2}{x+1} \, dx$.

  2. Mastering Techniques:

    • Develop skills sufficient to manage and simplify complex fractions for integration using partial fractions.

Worksheet 5: Improper Integrals

  1. Convergence/Divergence Tests:

    • Use comparison test to evaluate integrals across defined limits and ascertain if integrals converge.

    • Examples can include boundaries at infinity or points of discontinuity.

Integration Techniques Summary**

  1. Various Techniques:

    • Improper, definite, and indefinite integrals are part of preparatory work in calculus.

    • Incorporate trigonometric identities in all evaluations and mastering integrations techniques is paramount for success.

Area between Curves

  1. Sketching Areas:

  2. Volume of Solids Revolution:

  3. Arc Length Evaluations:

Conclusion

  • Mastering these varied worksheets builds a comprehensive foundation in calculus.

  • Emphasis on practice is essential in preparing for complex and multifaceted calculus problems pertaining to real applications.