STEM300-Lesson-8 (1)

Antiderivative Rules / Basic Integration

Lesson Overview

• Fundamental topic in calculus dealing with reversing the derivative process.

• The process of finding antiderivatives is called integration.

Antidifferentiation

• Antidifferentiation is the process of finding the antiderivatives of functions, which is the opposite process of differentiation.

• Example:

  • Given: 𝑦 = 4𝑥³ + 5𝑥² + 6

  • Derivative: 𝑦′ = 12𝑥² + 10𝑥

  • Antiderivative: 𝑦 = ∫(12𝑥² + 10𝑥)𝑑𝑥 = 4𝑥³ + 5𝑥² + 𝐶

Integration Steps

• Integrate each term separately:

  • ∫12𝑥²𝑑𝑥 = 4𝑥³ + 𝐶

  • ∫10𝑥𝑑𝑥 = 5𝑥² + 𝐶

Basic Integration Formulas

• Important rules used in integration involving various functions:

  1. Sum Rule: ∫(𝑓(𝑥) + 𝑔(𝑥))𝑑𝑥 = ∫𝑓(𝑥)𝑑𝑥 + ∫𝑔(𝑥)𝑑𝑥

  2. Constant Multiple Rule: ∫𝑐𝑓(𝑥)𝑑𝑥 = c∫𝑓(𝑥)𝑑𝑥

  3. Power Rule: ∫𝑥ⁿ𝑑𝑥 = (𝑥ⁿ⁺¹)/(𝑛 + 1) + 𝐶 , (where n ≠ -1)

  4. Trigonometric Functions:

     • ∫cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶

     • ∫sin 𝑥 𝑑𝑥 = -cos 𝑥 + 𝐶

     • ∫tan 𝑥 𝑑𝑥 = -ln|cos 𝑥| + 𝐶

     • ∫sec² 𝑥 𝑑𝑥 = tan 𝑥 + 𝐶

     • ∫csc² 𝑥 𝑑𝑥 = -cot 𝑥 + 𝐶

     • ∫csc 𝑥 cot 𝑥 𝑑𝑥 = -csc 𝑥 + 𝐶

Examples of Integration

• Demonstrating the use of integration:

  1. ∫4𝑥³𝑑𝑥 = (4/4)𝑥⁴ + 𝐶 = 𝑥⁴ + 𝐶

  2. ∫𝑥⁴𝑑𝑥 = (1/5)𝑥⁵ + 𝐶

  3. ∫7𝑥²𝑑𝑥 = (7/3)𝑥³ + 𝐶

Additional Examples

• Different cases demonstrating integration techniques:

  1. ∫1/𝑥² 𝑑𝑥 = -1/𝑥 + 𝐶

  2. ∫3𝑥³ 𝑑𝑥 = (3/4)𝑥⁴ + 𝐶

Other Integration Techniques

• Integration by substitution and trigonometric identities can simplify more complex integrals.

• Problems and specific functions can often require unique approaches depending on their forms.

Conclusion

• Integration allows for the recovery of the original function from its derivative.

• Mastery of integral formulas and techniques is critical for progression in calculus.

Acknowledgments

• Prepared by: Ms. Marialisa E. Virador

• Appreciation for participation in learning.