Integrating Cosine to the Fourth Power

Power Reduction Formula Application

• Objective: Integrate ( \cos^4(5x) )
• First Step: Convert ( \cos^4(5x) ) to a power of 2: ( (\cos^2(5x))^2 )

Applying Power Reduction Formula for ( \cos^2 )

• Rewrite using power reduction: ( \cos^2(5x) = \frac{1 + \cos(2\theta)}{2} )
• For ( \theta = 5x ), we get:
  • ( \cos^2(5x) = \frac{1 + \cos(10x)}{2} ) squared
  • Integrating this expression requires simplifying further

Simplifying the Expression

1. Expand: ( (1 + \cos(10x))^2 / 4
2. Simplification Steps
   • Numerator: Expand ( (1 + \cos(10x))^2 ) using FOIL (First, Outer, Inner, Last)
   • Denominator: Combine constants, in this case, squaring 2 to get 4

Expansion Math

• FOIL Expansion:
  • First: ( 1*1 = 1 )
  • Outer: ( 1 * \cos(10x) = \cos(10x) )
  • Inner: ( 1 * \cos(10x) = \cos(10x) )
  • Last: ( \cos(10x) * \cos(10x) = \cos^2(10x) )
  • Combine: ( 1 + 2\cos(10x) + \cos^2(10x) )

Integration Step-by-Step

1. Separate Terms:
   • Integrate: ( \frac{1}{4}(1 + 2\cos(10x) + \cos^2(10x)) )
2. Integrate Each Term:
   • ( \int 1 , dx = x )
     • Result: ( x / 4 )
   • ( \int 2\cos(10x) , dx )
     • Antiderivative of ( \cos(10x) ) is ( \frac{\sin(10x)}{10} )
     • Result: ( \frac{\sin(10x)}{20} )
   • ( \int \cos^2(10x) , dx )
     • Apply another power reduction formula:
       • ( \frac{1 + \cos(20x)}{2} , dx )
     • Combine constants: ( \frac{1}{8} \int 1 , dx + \frac{1}{8} \int \cos(20x) , dx )
     • Result: ( \frac{x}{8} + \frac{\sin(20x)}{160} )

Combining Results

• Final Integral:
  • ( \frac{x}{4} + \frac{\sin(10x)}{20} + \frac{x}{8} + \frac{\sin(20x)}{160} + C )
  • Simplify: ( \frac{3x}{8} + \frac{\sin(10x)}{20} + \frac{\sin(20x)}{160} + C )

Conclusion

• Result: Successfully integrated ( \cos^4(5x) )
• Remember integrated terms and constant of integration (C)
• Keep practicing power reduction and integral separation techniques