Analysis 2 Notes 6

Question 1

(6D-N48-6.1) Let f ∈ Rloc(I), I = ⟨c, d⟩. f integrable at c, d in the improper sense. f integrable over I. Know notes after too.

Answer 1

-

Question 2

(6E-N49-6.1) Evaluate definite integral 0 to 1 of x^a

Answer 2

-

Question 3

(6E-N50-6.2) Evalue (definite) integral 1 to plus infinity x^a

Answer 3

-

Question 4

(6T-N50-6.1) Let f : I → [0, +∞) be a locally Riemann integrable function. Then f is integrable over I iff

Answer 4

-

Question 5

(6T-N52-6.2) Comparison Theorem for Improper Integrals

Answer 5

-

Question 6

(6T-N52-6.3) Suppose that f ∈ Rloc(I) and |f| is integrable over I. Then f is integral over I

Answer 6

-

Question 7

(6E-N53-6.3) Prove existence of improper integral from 0 to plus infinity cos(x)/ (1 + x^a)

Answer 7

-

Question 8

(6E-N54-6.4) Prove existence of improper integral from 0 to plus infinity sin (x)/ x

Answer 8

-

Question 9

(6R-N54-) Improper integral 0 to plus infinity sin (x)/ x has value pi/2. Called Dirichlet integral

Answer 9

-

Question 10

(6R-N55-) Improper integral minus to plus infinity e^(-x^2) has value root pi

Answer 10

-

Question 11

(6T-N55-6.4) Let r be a natural number and let f : [r,+∞) → [0,+∞) be a decreasing function. Then infinite series f(k) from k=r converges iff..

Answer 11

-