Convergence and Divergence of Improper Integrals
- 1. Gandhinagar Institute of Technology(012) Active Learning Assignment Subject- Calculus(2110014) Topic- Convergence and Divergence of Improper Integrals Branch-Computer Engineering C : C-2 Prepared By- Somai Rohankumar J.(170120107169) Guided By- Prof. Mansi Vaishnani
- 2. Agenda • Direct Comparison Test • Example • Limit Comparison Test • Example
- 3. Direct Comparison Test (a) If f(x) and g(x) are two continuous functions for x ≥ a such that 0 ≤ f(x) ≤ g(x) then, 𝑎 ∞ 𝑓(𝑥) 𝑑𝑥 converges, if 𝑎 ∞ 𝑔(𝑥) 𝑑𝑥 converges and 𝑎 ∞ 𝑓 𝑥 𝑑𝑥 ≤ 𝑎 ∞ 𝑔(𝑥) 𝑑𝑥.
- 4. Direct Comparison Test (contd.) (b) If f(x) and g(x) are two continuous functions for x ≥ a such that f(x) ≥ g(x) then, 𝑎 ∞ 𝑓(𝑥) 𝑑𝑥 diverges , if 𝑎 ∞ 𝑔(𝑥) 𝑑𝑥 diverges.
- 5. Example • Test the convergence of the improper integral 1 ∞ 𝑐𝑜𝑠𝑥 𝑥2 𝑑𝑥 . • Solution
- 8. Limit Comparison Test • If f(x) and g(x) are two continuous functions for x ≥ a such that f(x) > 0 , g(x) > 0 and lim 𝑥→∞ 𝑓(𝑥) 𝑔(𝑥) = l , 0 < l < ∞ then 𝑎 ∞ 𝑓 𝑥 𝑑𝑥 and 𝑎 ∞ 𝑔 𝑥 𝑑𝑥 both converge or diverge simultaneously.
- 9. Example • Test the convergence of the improper integral 4 ∞ 3𝑥+5 𝑥4+7 𝑑𝑥 . • Solution
- 10. Example (contd.)
- 11. Example (contd.)
- 12. Example (contd.)
- 13. References • Calculus Textbook Published by Mc Graw Hill Eduation Written by Ravish R Singh & Mukul Bhatt