MATH 409 Advned Clulus I Leture 22: Improper Riemnn integrls.
Improper Riemnn integrl If funtion f : [,b] R is integrble on [,b], then the funtion F(x) = x f(t)dt is well defined nd ontinuous on [,b]. In prtiulr, F() F(b) s b, i.e., f(x)dx = lim b+ f(x)dx. Now suppose tht f is defined on the semi-open intervl J = [,b) nd is integrble on ny losed intervl [,d] J (suh funtion is lled lolly integrble on J). Then ll integrls in the right-hnd side re well defined nd the limit might exist even if f is not integrble on [,b]. If this is the se, then f is lled improperly integrble on J nd the limit is lled the (improper) integrl of f on [,b). Similrly, one defines improper integrbility on the semi-open intervl (,b].
Suppose funtion f is lolly integrble on semi-open intervl J = [, b) or (, b]. Then there re two possible obstrutions for f to be integrble on [, b]: (i) the funtion f is not bounded on J, nd (ii) the intervl J is not bounded. Exmples. Funtion f(x) = / x is improperly integrble on (0,]. 0 x dx = lim 0+ x dx = lim 0+ 2 x x= = lim 0+ (2 2 ) = 2. Funtion g(x) = x 2 is improperly integrble on [, ). x 2 dx = lim + = lim + ( ) =. x 2 dx = lim + x x=
Properties of improper integrls Sine n improper Riemnn integrl is limit of proper integrls, the properties of improper integrls re nlogous to those of proper integrls (nd derived using limit theorems). Theorem Let f : [,b) R be funtion integrble on ny losed intervl [,b ] [,b). Given (,b), the funtion f is improperly integrble on [,b) if nd only if it is improperly integrble on [, b). In the se of integrbility, f(x)dx = f(x)dx + f(x)dx. Sketh of the proof: For ny d (,b) we hve the following equlity involving proper Riemnn integrls: d f(x)dx = f(x)dx + d f(x)dx. The theorem is proved by tking the limit s d b.
Theorem Suppose tht funtion f : (,b) R is integrble on ny losed intervl [,d] (,b). Given number I R, the following onditions re equivlent: (i) for some (, b) the funtion f is improperly integrble on (,] nd [,b), nd f(x)dx + f(x)dx = I; (ii) for every (, b) the funtion f is improperly integrble on (,] nd [,b), nd f(x)dx + f(x)dx = I; (iii) for every (,b) the funtion f is improperly integrble on (,] nd f(x)dx I s b ; (iv) for every (,b) the funtion f is improperly integrble on [,b) nd f(x)dx I s +.
Improper integrl: two singulr points Definition. A funtion f : (, b) R is lled improperly integrble on the open intervl (, b) if for some (nd then for ny) (, b) it is improperly integrble on semi-open intervls (, ] nd [, b). The integrl of f is defined by f(x)dx = f(x)dx + f(x)dx. In view of the previous theorem, the integrl does not depend on. It n lso be omputed s repeted limit: ( d ) ( d ) f(x)dx = lim lim f(x)dx = lim lim f(x)dx. d b + + d b Finlly, the integrl n be omputed s double limit (i.e., the limit of funtion of two vribles): f(x)dx = lim + d b d f(x)dx.
More properties of improper integrls If funtion f is integrble on losed intervl [,b] or improperly integrble on one of the semi-open intervls [, b) nd (, b], then it is lso improperly integrble on the open intervl (, b) with the sme vlue of the integrl. If funtions f, g re improperly integrble on (, b), then for ny α,β R the liner ombintion αf +βg is lso improperly integrble on (, b) nd ( ) b αf(x)+βg(x) dx = α f(x)dx +β g(x)dx. Suppose funtion f : (,b) R is lolly integrble nd hs n ntiderivtive F. Then f is improperly integrble on (,b) if nd only if F(x) hs finite limits s x + nd s x b, in whih se f(x)dx = lim F(x) lim F(x). x b x +
Comprison Theorems for improper integrls Theorem Suppose tht funtions f, g re improperly integrble on (,b). If f(x) g(x) for ll x (,b), then f(x)dx g(x)dx. Theorem 2 Suppose tht funtions f, g re lolly integrble on (, b). If the funtion g is improperly integrble on (,b) nd 0 f(x) g(x) for ll x (,b), then f is lso improperly integrble on (, b). Theorem 3 Suppose tht funtions f,g,h re lolly integrble on (, b). If the funtions g, h re improperly integrble on (,b) nd h(x) f(x) g(x) for ll x (,b), then f is lso improperly integrble on (,b) nd h(x)dx f(x)dx g(x)dx.
Exmples Funtion f(x) = x 2 is not improperly integrble on (0, ). Indeed, the ntiderivtive of the funtion f, whih is F(x) = x, hs finite limit s x + but diverges to infinity s x 0+. Funtion g(x) = x 2 osx is improperly integrble on [, ). We hve f(x) g(x) f(x) = x 2 for ll x. Sine the funtion f is improperly integrble on [, ), it follows tht f is lso improperly integrble on [, ). By the Comprison Theorem for improper integrls, the funtion g is improperly integrble on [, ) s well.
Exmples Funtion f(x) = e x is improperly integrble on [0, ). Indeed, the ntiderivtive of the funtion f, whih is F(x) = e x, hs finite limit s x +. Funtion g(x) = e x2 is improperly integrble on (, ). We hve 0 g(x) f(x) = e x for ll x. Sine the funtion f is improperly integrble on [0, ), it follows tht g is improperly integrble on [, ). Sine the funtion g is even, g( x) = g(x), it follows tht g is lso improperly integrble on (, ]. Finlly, g is properly integrble on [,].
Exmples Funtion f(x) = x sinx is improperly integrble on [, ). To show improper integrbility, we integrte by prts: x sinx dx = x d(osx) = x osx + osx d(x ) x= = os os x 2 osx dx. Sine the funtion g(x) = x 2 osx is improperly integrble on [, ) nd os 0 s +, it follows tht f is improperly integrble on [, ).
Absolute integrbility Definition. A funtion f : (, b) R is lled bsolutely integrble on (, b) if f is lolly integrble on (, b) nd the funtion f is improperly integrble on (, b). Theorem If funtion f is bsolutely integrble on (, b), then it is lso improperly integrble on (, b) nd f(x)dx f(x) dx. Proof: Sine f is improperly integrble on (, b), so is f. Clerly, f(x) f(x) f(x) for ll x (,b). By the Comprison Theorems for improper integrls, the funtion f is improperly integrble on (, b) nd f(x) dx f(x)dx f(x) dx.
Exmples For ny nonnegtive funtion, the bsolute integrbility is equivlent to improper integrbility. In prtiulr, the funtion f (x) = x 2 is bsolutely integrble on [, ) nd is not on (0, ). The funtion f 2 (x) = / x is bsolutely integrble on (0,). The funtion f 3 (x) = e x2 is bsolutely integrble on (, ). Funtion f(x) = e x2 sinx is bsolutely integrble on (, ). Indeed, the funtion f is lolly integrble on (, ), funtion g(x) = e x2 is improperly integrble on (, ), nd f(x) g(x) for ll x R.
Counterexmples { if x Q, Funtion f(x) = if x R\Q bsolutely integrble on (0, ). is not Indeed, the funtion f is not lolly integrble on (0, ). At the sme time, the funtion f is onstnt nd hene (properly) integrble on (0, ). Funtion f(x) = x sinx is not bsolutely integrble on [, ). (n+)π (n+)π sinx For ny n N, f(x) dx nπ nπ (n+)π dx π 2 = sinx dx = (n+)π 0 (n+)π nπ (n+)π dx π nπ x. It remins to notie tht g(x) = /x is not improperly integrble on [π, ).