Infi-2 9

Improper integral of the second type

What if interval is fine, but the function is not bounded on it?For example, ∫011xdx

Integrability on non-closed interval #definition

Function f is called Riemann-integrable on (a,b]If ∀c∈(a,b]:f is Riemann-integrable on [c,b]∫abf(x)dx=limc→a+∫cbf(x)dxFunction f is called Riemann-integrable on [a,b)If ∀c∈[a,b):f is Riemann-integrable on [a,c]∫abf(x)dx=limc→b−∫acf(x)dx
∫01/21xln⁡xdx=limc→0+∫c1/21xln⁡xdx==limc→0+ln⁡|ln⁡x||c1/2=ln⁡|ln⁡(12)|−ln⁡|ln⁡c|=−∞⟹Integral diverges
If integral has multiple "problems", or they are in the middle of the interval,integral is to be calculated as a sum of integralsIntegral then converges iff each additive converges
Complex integrals might need a lot of sub-intervals:∫−∞∞1(x−6)(x−23)dx==∫−∞0+∫06+∫68+∫823+∫2324+∫24∞

Comparison tests

p-Integral test for improper integrals of the first type #lemma

∫ab1(x−a)pdx converges⟺p<1

Comparison test for improper integrals of the second type #lemma

Let f,g be Riemann-integrable on (a,b]Let 0≤f≤gThen ∫abg(x)dx converges⟹∫abf(x)dx converges

Limit comparison text for improper integrals of the second type #lemma

Let f,g be Riemann-integrable on (a,b]Let 0≤f,gL=limx→a+f(x)g(x)L=∞⟹[∫abf(x)dx converges⟹∫abg(x)dx converges]L=0⟹[∫abf(x)dx converges⟸∫abg(x)dx converges]0<L<∞⟹[∫abf(x)dx converges⟺∫abg(x)dx converges]