Calculus Lecture Notes: Integration and Anti-Derivatives

Key Topic: Finding the Anti-Derivative

Introduction

• Objective: Find the anti-derivative of a given function or evaluate the integral.
• Example Function: Integral of ( x^5 \cos(2x^6) \ dx ).

Step-by-Step Solution

1. Factoring Constant Out

• Factor out the constant 9 from the integral.

2. U-Substitution

• Let ( U = x^6 ).
• Therefore, ( dU = 6x^5 \ dx ).
• Rearrange to find ( x^5 dx ):
  • ( \frac{1}{6} dU = x^5 dx ).

3. Integrating with U

• Substitute ( x^5 dx ) with ( \frac{1}{6} dU ).
• Integral becomes: ( 9 \cdot \frac{1}{6} \int \cos(U) , dU ).

4. Cosine Power Reduction

• No direct integration formula for ( \cos(U) ).
• Apply power-reducing formula:
  • ( \cos^2(U) = \frac{1}{2} (1 + \cos(2U)) ).

5. Applying Power Reduction

• Factor out constants to simplify:
  • ( \frac{3}{2} \cdot \int (1 + \cos(2U)) , dU ).

6. Finding the Anti-Derivative

• Integrate:
  • ( \int 1 , dU = U ).
  • ( \int \cos(2U) , dU ) requires substitution.

7. V-Substitution for Cosine

• Let ( V = 2U ).
• Then ( dV = 2 , dU ) or ( dU = \frac{1}{2} , dV ).
• Integrate ( \cos(V) , dV ) gives ( \sin(V) ).

8. Combine Results

• ( \frac{3}{4} U + \frac{3}{8} \sin(2U) + C ).

9. Substituting Back to X

• Substitute ( U = x^6 ) back:
  • Final anti-derivative:
  • ( \frac{3}{4} x^6 + \frac{3}{8} \sin(2x^6) + C ).

Conclusion

• Steps provide a structured approach to evaluating integrals with substitution and power reduction techniques.
• Important to track and manage substitutions correctly for accurate integration.