Indefinite Integral Problems Lecture Notes

Basic Principles

• Anti-derivative of a Constant:
  • Integral of 4 dx: Add an 'x' to the constant. Result: 4x + c
  • Example: Integral of π dy: Result: πy + c
• Integral of a Constant with Respect to Another Variable:
  • Integral of e dz: Result: ez + c

Integrals of Powers of Variables

• Formula for Powers:
  • Integral of x^n dx: Result: x^(n+1)/(n+1) + c
  • Example: Integral of x^2 dx: Result: x^3/3 + c
• Integrals Involving Coefficients:
  • Example: Integral of 8x^3 dx: Result: 2x^4 + c

Polynomial Functions

• Integrate Terms Separately:
  • Example: Integral of x^2 - 5x + 6 dx: Result: x^3/3 - 5x^2/2 + 6x + c

Fractions and Roots

• Square Roots:
  • Rewriting square root functions for integration, e.g., Integral of √x dx: Rewrite as x^(1/2), Result: 2/3 x^(3/2) + c
• Fractional Powers:
  • Example: Integral of x^(4/3) dx: Result: 3/7 x^(7/3) + c

Special Cases

• Reciprocal Functions:
  • Integral of 1/x dx: Result: ln|x| + c
• Exponential Functions:
  • Example: Integral of e^(4x) dx: Result: e^(4x)/4 + c

Trigonometric Functions

• Basic Trigonometric Integrals:
  • Integral of cos(x) dx: Result: sin(x) + c
  • Integral of sin(x) dx: Result: -cos(x) + c
• Advanced Trigonometric Integrals:
  • Integral involving sec²(x) or sec(x)tan(x).

Techniques for Complex Expressions

• U Substitution:
  • Useful for expressions like x²sin(x³) dx by substituting u = x³.
• Integration by Parts:
  • Formula: ∫u dv = uv - ∫v du.
  • Example: Integral of x cos(x) dx.

Additional Techniques

• Trigonometric Substitution:
  • Used for expressions like 4/(1 + x²) dx, replacing x with tangent(θ).

Conclusion

• Practice with each type of function to reinforce understanding.
• Use substitutions and transformations to simplify more complex integrals.