Lecture Notes: Evaluating Improper Integrals

Introduction to Improper Integrals

• Definition: An integral with infinity as a limit (either upper or lower) is called an improper integral.
  • Convergent: The integral yields a finite number.
  • Divergent: The integral yields infinity.
• Example: Integral from 1 to infinity of 1/x dx

Evaluating the Integral ( \int_{1}^{\infty} \frac{1}{x} dx )_

1. Express as a Limit:
   • Replace infinity with a variable (t).
   • Expression: ( \lim_{t \to \infty} \int_{1}^{t} \frac{1}{x} dx )
2. Antiderivative:
   • ( \int \frac{1}{x} dx = \ln |x| )
3. Apply Fundamental Theorem of Calculus:
   • ( \lim_{t \to \infty} [\ln(t) - \ln(1)] )
   • ( \ln(1) = 0 )
   • Evaluate: ( \lim_{t \to \infty} \ln(t) = \infty )
   • Conclusion: Divergent (yields infinity).

Example: Integral ( \int_{1}^{\infty} \frac{1}{x^2} dx )_

1. Express as a Limit:
   • ( \lim_{t \to \infty} \int_{1}^{t} \frac{1}{x^2} dx )
2. Antiderivative:
   • Rewrite: ( x^{-2} )
   • Apply Power Rule: Add 1 to exponent, divide by new exponent.
   • ( \int x^{-2} dx = -\frac{1}{x} + C )
3. Calculate:
   • ( \lim_{t \to \infty} [-\frac{1}{t} - (-1)] )
   • Evaluate: ( \lim_{t \to \infty} -\frac{1}{t} = 0 )
   • Result: 1 (finite number)
   • Conclusion: Convergent

P-Series Test

• Rule: ( \int_{1}^{\infty} \frac{1}{x^p} dx )
  • Convergent if ( p > 1 )
  • Divergent if ( p \leq 1 )
• Examples:
  • ( p = 1 ) (Divergent)
  • ( p = 2 ) (Convergent)_

Additional Example: ( \int_{1}^{\infty} \frac{1}{(3x+1)^2} dx )_

1. Comparison:
   • Similar to ( \int \frac{1}{x^2} dx ), where ( p = 2 ), thus convergent by comparison.
2. Verification:
   • Use substitution: Let ( u = 3x + 1 ), ( du = 3 dx )
   • New integral: ( \int \frac{1}{u^2} \cdot \frac{1}{3} du )
   • Antiderivative: ( -\frac{1}{3u} )
3. Compute:
   • Substitute back ( u = 3x + 1 )
   • ( \lim_{t \to \infty} \left[ -\frac{1}{3(3t+1)} + \frac{1}{12} \right] )
   • Evaluate: Limit approaches zero, thus final result is ( \frac{1}{12} )
   • Conclusion: Convergent_