Improper Integrals. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics


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1 Improper Integrls MATH 2, Clculus II J. Robert Buchnn Deprtment of Mthemtics Spring 28
2 Definite Integrls Theorem (Fundmentl Theorem of Clculus (Prt I)) If f is continuous on [, b] then b f (x) dx = [F(x)] x=b x= = F(b) F() where F is ny ntiderivtive of f on (, b).
3 Definite Integrls Theorem (Fundmentl Theorem of Clculus (Prt I)) If f is continuous on [, b] then b f (x) dx = [F(x)] x=b x= = F(b) F() where F is ny ntiderivtive of f on (, b). Question: Cn we evlute the definite integrl x 2 dx?
4 Answer We cnnot use the Fundmentl Theorem of Clculus to evlute x 2 dx since the integrnd hs discontinuity t x =. If we try to evlute it using the Fundmentl Theorem of Clculus we get [ x 2 dx = ] x= = 2 x x= result which is impossible since /x 2 > for x < nd < x.
5 Improper Integrls Extr cre must be exercised when ttempting to evlute definite integrls for which the intervl over which we integrte is of infinite length (Type ),
6 Improper Integrls Extr cre must be exercised when ttempting to evlute definite integrls for which the intervl over which we integrte is of infinite length (Type ), nd/or the integrnd possesses isolted discontinuities within the integrtion intervl (Type 2).
7 First Type (Type ) Definition If f is continuous on [, ) we define the improper integrl R f (x) dx f (x) dx. R If f is continuous on (, ] we define the improper integrl f (x) dx = lim R R f (x) dx. If the limit is L (finite) we sy the improper integrl converges, otherwise we sy it diverges.
8 Exmples Determine if the following improper integrls converge or diverge e x dx x dx x 2 dx
9 e x dx e x dx R This improper integrl converges. R e x dx [ e x ] x=r R x= [ e R + e ] R = e
10 5 x dx 5 dx x = R lim R 5 x dx [ln x ]x=r x=5 R [ln R ln 5] = This improper integrl diverges. R
11 5 x 2 dx 5 dx x 2 R R R R 5 [ x x 2 dx ] x=r x=5 [ R + 5 ] = 5 This improper integrl converges.
12 Intervl (, ) Definition If f is continuous on (, ) then f (x) dx = for ny constnt. We sy diverges. f (x) dx nd f (x) dx + f (x) dx f (x) dx converges if both f (x) dx converge, otherwise f (x) dx
13 Exmple Determine if the following improper integrl converges. e x dx + e x
14 Solution ( of 2) = e x + e x dx e x + e R R R R dx + x e x + e e x + e x dx + lim e x e 2x dx + lim + S S S dx x S e x dx + e x e x e 2x + dx Use integrtion by substitution with u = e x nd du = e x dx.
15 Solution (2 of 2) e x + e R R = R dx x e R [ π 4 ] + e S u 2 du + lim + S u 2 + du [ ] u= [ ] u=e tn u + lim tn S u u=e R S u= [ tn tn e R] + lim S [ π 2 π ] = π 4 2 This improper integrl converges. [ tn e S tn ]
16 Grphicl Approch Suppose f nd g re two continuous functions defined on [, ) nd such tht f (x) g(x) for ll x. y f(x) g(x) x
17 Grphicl Approch Suppose f nd g re two continuous functions defined on [, ) nd such tht f (x) g(x) for ll x. y f(x) g(x) x If f (x) dx diverges, wht bout g(x) dx?
18 Grphicl Approch Suppose f nd g re two continuous functions defined on [, ) nd such tht f (x) g(x) for ll x. y f(x) g(x) x If If f (x) dx diverges, wht bout g(x) dx converges, wht bout g(x) dx? f (x) dx?
19 Comprison Test Theorem (Comprison Test) Suppose tht f nd g re continuous on [, ) nd f (x) g(x) for ll x.. If 2. If g(x) dx converges, then f (x) dx diverges, then f (x) dx converges. g(x) dx diverges.
20 Exmples Determine if the following improper integrls converge or diverge x 2 2 x dx + sec 2 x x e x+ dx dx
21 x 2 2 x dx Note tht so x 2 2 x < x 2 x < x 2 x 4 = x 2 x 2 2 x dx < x 2 dx R R x 2 dx [ ] x=r R x x= [ R ] + =. R The originl improper integrl converges.
22 + sec 2 x x dx Note tht so + sec 2 x x > x + sec 2 x x dx > R x dx R x dx [ln x ]x=r x= R [ln R ln ] =. R The originl improper integrl diverges.
23 e x+ dx Note tht so e x+ dx > e x+ = e e x > e x R e x dx R e x dx R [ex ] x=r x= [ ] e R =. R The originl improper integrl diverges.
24 Second Type (Type 2) Definition If f is continuous on the intervl [, b) nd f (x) s x b, the improper integrl of f on [, b] is b f (x) dx = R lim f (x) dx. R b If f is continuous on the intervl (, b] nd f (x) s x +, the improper integrl of f on [, b] is b f (x) dx = b lim f (x) dx. R + R If the limit is L (finite), we sy the improper integrl converges, otherwise we sy it diverges.
25 Exmples Determine if the following improper integrls converge or diverge π/2 2x x 2 dx 3 x dx tn x dx
26 4 2x x 2 dx 4 2x 4 2x x 2 dx R + R x 2 dx [ ] x=4 ln x 2 R + x=r ] ln 5 ln R 2 = R + [ This improper integrl diverges.
27 3 x dx 3 dx x /3 dx x R + R [ ] 3 x= R + 2 x 2/3 x=r [ 3 R ] 2 R2/3 = 3 2 This improper integrl converges.
28 π/2 tn x dx π/2 tn x dx R R π/2 R R π/2 This improper integrl diverges. tn x dx sin x cos x dx [ ln cos x ]x=r R π/2 x= [ ln cos R + ln cos ] R π/2 ( ) = lim ln cos R R π/2 =
29 Discontinuity in (, b) Definition Suppose f is continuous on [, b] except t some c (, b) nd f (x) s x c. The improper integrl is If c b b f (x) dx = f (x) dx = L nd b c c b f (x) dx + f (x) dx. c f (x) dx = L 2 the improper integrl f (x) dx converges to L + L 2. If either of the improper integrls c diverges s well. f (x) dx or b c f (x) dx diverges then b f (x) dx
30 Exmples Determine if the following improper integrls converge or diverge x dx 2x x 2 dx
31 3 x dx 3 x dx = R R R 3 dx + x R 3 x dx x /3 dx + lim [ ] 3 x=r 2 x 2/3 [ 3 2 R2/3 3 2 x= S + + lim S + ] + lim S + S x /3 dx [ ] 3 x= 2 x 2/3 x=s [ S2/3 ] = = This improper integrl converges.
32 4 2x x 2 dx 4 2x x 2 dx = 2x 4 x 2 dx + 2x x 2 dx We hve lredy shown tht the second integrl on the righthnd side of the eqution diverges, thus this improper integrl diverges.
33 Homework Red Section 7.8 Exercises: WebAssign/D2L
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