U-Substitution for Definite Integrals

Key Concepts

• U-substitution is a method used to find antiderivatives, particularly useful for definite and indefinite integrals.
• The goal is to change variables from x to u to simplify integration.
• Steps to follow:
  1. Identify u and du.
  2. Solve for dx in terms of du.
  3. Substitute back into the integral, cancel terms if possible.
  4. Integrate using the new variable u.
  5. Substitute back the original variable.

Example Problems

Problem 1: ( \int 4x(x^2 + 5)^3 dx )

• Choose: ( u = x^2 + 5 )
  • ( du = 2x , dx )
  • Solve for ( dx: ) [ dx = \frac{du}{2x} ]
• Substitute and simplify:
  • ( \int 4x u^3 \frac{du}{2x} = 2 \int u^3 du )
• Integrate:
  • ( 2 \times \frac{u^4}{4} + C = \frac{1}{2} u^4 + C )
• Substitute back:
  • Final Answer: ( \frac{1}{2} (x^2 + 5)^4 + C )

Problem 2: ( \int 8\cos(4x) , dx )

• Choose: ( u = 4x )
  • ( du = 4 , dx )
  • Solve for ( dx: ) [ dx = \frac{du}{4} ]
• Substitute and simplify:
  • ( \int 8\cos(u) \frac{du}{4} = 2 \int \cos(u) , du )
• Integrate:
  • ( 2\sin(u) + C )
• Substitute back:
  • Final Answer: ( 2\sin(4x) + C )

Problem 3: ( \int x^3 e^{x^4} , dx )

• Choose: ( u = x^4 )
  • ( du = 4x^3 , dx )
  • Solve for ( dx: ) [ dx = \frac{du}{4x^3} ]
• Substitute and simplify:
  • ( \int e^u \frac{du}{4} = \frac{1}{4} \int e^u , du )
• Integrate:
  • ( \frac{1}{4}e^u + C )
• Substitute back:
  • Final Answer: ( \frac{1}{4} e^{x^4} + C )

Problem 4: ( \int 8x \sqrt{40 - 2x^2} , dx )

• Choose: ( u = 40 - 2x^2 )
  • ( du = -4x , dx )
  • Solve for ( dx: ) [ dx = \frac{du}{-4x} ]
• Substitute and simplify:
  • ( \int 8x \sqrt{u} \frac{du}{-4x} = -2 \int u^{1/2} , du )
• Integrate:
  • ( -\frac{4}{3} u^{3/2} + C )
• Substitute back:
  • Final Answer: ( -\frac{4}{3} (40 - 2x^2)^{3/2} + C )

Strategies

• Choose u such that substitution simplifies the integral.
• Isolate dx to avoid mistakes.
• Replace all x variables with u variables.
• Cancel terms to simplify.
• Use power rule for integration: ( \int u^n du = \frac{u^{n+1}}{n+1} + C )

Additional Practice

• Try integrating: ( \int x^3 (2 + x^4)^2 , dx )
• Consider trying similar problems to reinforce learning.

Conclusion

• U-substitution simplifies finding antiderivatives.
• Practice with various functions to become proficient.
• Always verify by differentiating the result to check correctness.