3 dx 10. Find Cauchy’s principal value of 1 x 4 2 11. Evaluate ln(sin ) x dx 0 12. that cos 2 xdx converges 0 (Hint: Make the substitution u = x2) a f(x)dx 13. Define - - 14. Test the following integrals for convergence sin x x b). dx a). 0 1 lnx 3 x x d). 2 dx dx c). 0 0 (1 x ) sinx  2 2 93 5 15. Test whether the principal value for the integral dx exists. 2 x 4 sin cos 4 x x dx 16. Show that converges. 0 x Summary You have learnt the following from the chapter: i. Definition of improper integrals of first and second kind. ii. Convergence and divergence of improper integrals. iii. Absolute convergence and conditional convergence of improper integrals. iv. Cauchy’s inequality for absolute convergence.
10. Find Cauchy’s principal value of 4 1 3 x dx 11. Evaluate 2 ln(sin 0 x ) dx 12. that cos 0 2 xdx converges (Hint: Make the substitution u = x2) a 13. Define f(x)dx - - 14. Test the following integrals for convergence a). 0 sin x b). 3 1 x lnx dx c). 0 x 2 (1  x 2 ) 2 dx d). 0 x sinx dx 93 15. Test whether the principal value for the integral 5 2 dx x 4 exists. 16. Show that 0 sin x cos 4 x x dx converges. Summary You have learnt the following from the chapter: i. Definition of improper integrals of first and second kind. ii. Convergence and divergence of improper integrals. iii. Absolute convergence and conditional convergence of improper integrals. iv. Cauchy’s inequality for absolute convergence.
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