Lecture Notes: Integration Using U-Substitution

Introduction to U-Substitution

• Focus on definite integrals
• Aim: Find the anti-derivative of complex expressions using substitution.

Key Concepts

• U-Substitution: A method to simplify integration by substituting part of the integrand with a single variable, u.
• Identifying U and DU:
  • u is chosen such that its derivative, du, simplifies the integral.
  • All x terms must be converted into terms of u.

Example Problems

Problem 1: 4x * (x^2 + 5)^3

1. Choose u:
   • u = x^2 + 5
   • du = 2x dx
   • Solve for dx: dx = du / (2x)
2. Substitute into integral:
   • Resulting expression: 4x * u^3 * (du / 2x)
   • Simplifies to: 2 * ∫u^3 du
3. Integrate:
   • 2 * (u^4 / 4) + C
   • Simplify: 1/2 * u^4 + C
4. Back-substitute:
   • Result: 1/2 * (x^2 + 5)^4 + C

Problem 2: 8 cos(4x) dx

1. Choose u:
   • u = 4x
   • du = 4 dx
   • Solve for dx: dx = du / 4
2. Substitute:
   • Resulting expression: 8 cos(u) * (du / 4)
   • Simplifies to: 2 ∫cos(u) du
3. Integrate:
   • Result: 2 sin(u) + C
4. Back-substitute:
   • Result: 2 sin(4x) + C

General Strategy for U-Substitution

• Identify u as a part of the integrand that, when derived, simplifies the integration.
• Express dx in terms of du and substitute in the integrand.
• Simplify the resulting integral and solve.
• Back-substitute the original terms for u.

Additional Practice Problems

Problem 3: x^3 e^(x^4)

• Choose u: u = x^4
• Solution Overview:
  • Simplify original problem using u and solve integral.
  • Final Answer: 1/4 * e^(x^4) + C

Problem 4: 8x * √(40 - 2x^2) dx

• Choose u: u = 40 - 2x^2
• Solution Overview:
  • Simplify using u, solve integral.
  • Final Answer: -4/3 * (40 - 2x^2)^(3/2) + C

Problem 5: x^3 / (2 + x^4)^2

• Choose u: u = 2 + x^4
• Solution Overview:
  • Simplify, solve integral.
  • Final Answer: -1/(4(2 + x^4)) + C

Problem 6: sin^4(x) * cos(x) dx

• Choose u: u = sin(x)
• Solution Overview:
  • Solve integral using u substitution.
  • Final Answer: 1/5 * sin^5(x) + C

Conclusion

• U-substitution is a versatile technique to handle a variety of integration problems by reducing complexity.
• Key is to correctly identify u and convert the entire integrand into terms of u.
• Practice is essential to mastering different scenarios and applications.

Additional Examples

• Includes more examples exploring other forms of functions, such as polynomials and trigonometric functions, to provide substantial practice.