Review-2 and Practice problems. sin 2 (x) cos 2 (x)(sin(x)dx) (1 cos 2 (x)) cos 2 (x)(sin(x)dx) let u = cos(x), du = sin(x)dx. = (1 u 2 )u 2 ( du)
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1 . Trigonometric Integrls. ( sin m (x cos n (x Cse-: m is odd let u cos(x Exmple: sin 3 (x cos (x Review- nd Prctice problems sin 3 (x cos (x Cse-: n is odd let u sin(x Exmple: cos 5 (x cos 5 (x sin (x cos (x(sin(x ( cos (x cos (x(sin(x let u cos(x, du sin(x ( u u ( du u 4 u du 5 u5 3 u3 + C 5 cos5 (x 3 cos3 (x + C (cos (x (cos(x ( sin (x (cos(x let u sin(x, du cos(x ( u ( du u 4 u + du u5 3 u3 + u + C sin5 (x 3 sin3 (x + sin(x + C Cse-3: m nd n re even use cos (x ( + cos(x nd sin( x ( cos(x Exmple: 6 sin 4 (x cos 4 (x
2 6 sin 4 (x cos 4 (x ( ( 6 ( + cos(x ( cos(x ( cos (x + cos 4 (x ( ( ( ( cos(4x + 4 ( + cos(4x + 4 cos(4x + 8 ( 3 8 cos(4x + 8 cos(8x ( ( + cos(4x ( + cos(4x + cos (4x ( + cos(8x 3 8 x 8 sin(4x + 64 sin(8x + C (b tn m (x sec n (x Cse-: n is even let u tn(x Exmple: tn (x sec 4 (x tn (x sec 4 (x tn (x sec (x(sec (x tn (x(tn (x + (sec (x let u tn(x, du sec (x u (u + (du u 4 + u du 5 u5 + 3 u3 + C 5 tn5 (x + 3 tn3 (x + C Cse-: m is odd let u sec(x Exmple: tn 3 (x sec(x
3 3 tn 3 (xsec(x (tn (x(tn(x sec(x (sec (x (tn(x sec(x let u sec(x, du tn(x sec(x (u ( du u du 6 u3 u + C 6 sec3 (x sec(x + C Cse-3: n is odd nd n is even No generl solution Exmple: sec(x sec(x(sec(x + tn(x sec(x sec(x + tn(x (sec (x + sec(x tn(x sec(x + tn(x let u sec(x + tn(x, du (sec (x + sec(x tn(x u ( du ln(u ln(sec(x + tn(x. Trigonometric substitution. Cse-: x let x sec(x Exmple: x 6x x 6x x 6x (x 3 3 let u x 3, du u 3 du let u 3 sec(θ, du 3 sec(θ tn(θdθ 3 sec(θ tn(θdθ 3 tn(θ
4 4 sec(θdθ ln sec(θ + tn(θ + C ln u 3 + u C ln x 3 + x 6x 3 + C ln x 3 + x 6x + C Cse-: x let x sin(x 4x Exmple: x 4x (x x 4x (x 4x ((x (x let u x, du u du let u sin(θ, du cos(θdθ cos(θ cos(θdθ 4 cos (θdθ 4 (cos(θ + dθ ( ( 4 sin(θ + θ + C sin(θ + θ + C sin(θ cos(θ + θ + C u u ( u + rcsin + C (x ( 4x x x + rcsin + C
5 5 Cse-3: x + let x tn(x Exmple: 4x + 9 4x + 9 (x + 3 let u x, du u + 3 du let u 3 tn(θ, du 3 sec (θdθ 6 sec(θ 3 sec (θdθ sec(θdθ ln sec(θ + tn(θ + C ln u u C ln 4x x 3 + C ln 4x x + C 3. Prtil frction. N(x Question: f(x, where N(x nd D(x re polynomils. D(x If degree of N(x degree of D(x Long division Prtil frction If degree of N(x < degree of D(x Prtil frction; Exmple-: (x (x + (x (x + A x + B x + Ax + A + Bx B (x (x + (A + Bx + (A B (x (x + Compre the coefficients, A + B nd A B, which leds to A nd B. (x (x + x + x +
6 6 ln x ln x + + C ln x x + + C Exmple-: To ensure x + 3x (x (x + x + 3x (x (x + A x + B x + + C (x + A(x + + B(x (x + + C(x (x (x + x + 3x A(x + + B(x (x + + C(x Tke x, 4 4A A ; Tke x, C C ; Tke x, A B C B ; x + 3x (x (x + x + x + + (x + ln x + ln x + + ( x + + C ln (x (x + x + + C Exmple-3: To ensure x + x + (x + (x + x + x + (x + (x + A x + + Bx + C x + A(x + + (Bx + C(x + (x + (x + x + x + A(x + + (Bx + C(x + Tke x, A A ; Tke x, A + C C ;
7 7 Tke x, 6 A + B + C B ; x + x + (x + (x + x + + x + x + ln x + + x x + + x + ln x + + ln(x + + rctn(x + C Exmple-3: x 4 + x 3 + 3x + x + x(x + x 4 + x 3 + 3x + x + x(x + A x + Bx + C x + + Dx + E (x + A(x + + (Bx + Cx(x + + (Dx + Ex (x + (x + (A + Bx4 + Cx 3 + (A + B + Dx + (C + Ex + A (x + (x + Constnt term, A ; x 3 term, C ; x term, C + E E ; x 4 term, A + B B ; x term, A + B + D 3 D ; x 4 + x 3 + 3x + x + x x 4 + x Exmple-4: x Long division: x + x + + x (x + ln x + rctn(x x + x x 4 + x 3 + x + x + x 4 + x 3 x x + x + x + x (x + + C
8 8 x 4 + x x x + + x 3 x3 + x + (x (x + follow Exmple- 3 x3 + x + ln x x + + C 4. Using integrl tble. (No generl procedures for problems in this section. Red lecture note of section.4 nd try to do exercise problems in textbook. 5. Numericl Integrtion. Trpezoidl Rule: x < x < < x n b, x i x i h b n I T (f(x h + f(x + + f(x n + f(x n Error Estimte for Trpezoidl Rule: Define E T E T K(b 3 n, b f(x I T, where K > nd f (x K for ll x in (, b. Simpson s Rule: x < x < < x n b, x i x i h b n I S (f(x 3 h + 4f(x + f(x + + f(x n + 4f(x n + f(x n Error Estimte for Simpson s Rule: Define E S E S where K > nd f (x K for ll x in (, b. Exmple: Estimte errors. Trpezoidl Rule: I T K(b 5 8n 4, b f(x I S, 5x 4 using Trpezoidl rule nd Simpson s rule nd estimte their h ( 4 ( f( + f( + f( + f( + f( 9
9 9 Error Estimte for Trpezoidl Rule: f (x 6x f (x < f ( 4 K, Simpson s Rule: I S 3 E T 4( ( ( f( + 4f( + f( + 4f( + f( Error Estimte for Simpson s Rule: f (x K 6. Improper integrls. Cse-.: Exmple: Cse-.: Exmple: + E S f(x lim t + xe x + f(x lim t x x + t ( ( / f(x xe x lim t + t xe x Integrtion by prts lim t + ( xe x e x t lim t + [( te t e t ( e e ] [( ( ] t f(x x x lim x + t t x + let u x +, du x lim t x xt u du lim [ln(u] x t xt lim t [ln(x + ] x xt lim [ln( t ln(t + ] x is divergent. x +
10 Cse-.3: Exmple: + + f(x x e x + f(x + + f(x x e x xe x + lim t t + xe x + lim t + let u x, du x lim t lim x xt t (e u x xt t (e x x xt lim e u du + lim t + xe x t xt x xt x xt + lim t + ( e u + lim t + ( e x xe x x lim t ( e t + lim t + ( e t + ( + ( + e u du Cse-.: If f(x is continuous in [, b but discontinuous t b, b f(x lim t b t f(x Exmple: x x + s x t lim x t x ( lim ( x t t lim t [ ( t ( ( ] [ + (] Cse-.: If f(x is continuous in (, b] but discontinuous t, b f(x lim t + b t f(x
11 Exmple: 3 9 x 9 x + s x 3+ 3 lim 9 x t x let x 3 sin(θ, 3 cos(θdθ lim t lim t cos(θ 3 cos(θdθ dθ lim θ x t 3 + xt [ ( x ] x lim rcsin t xt lim t 3 +[rcsin( rcsin(t/3] [rcsin( rcsin( ] ( π π Cse-.3: If f(x is continuous in [, c nd (c, b] but discontinuous t c, b f(x c f(x + b c f(x Exmple: x x + s x x lim t + + x t x x + lim t t x lim t +( ( x x + lim xt t (x xt x lim t +( ( ( t + lim t (t
12 ( ( + ( 4 Cse-.4: If f(x is continuous in (, b but discontinuous t nd b, b f(x where c could be ny point in (, b. Exmple: 4 x 4 x c f(x + b c f(x, + s x or 4 x + 4 x 4 x let x sin(θ, cos(θdθ cos(θ cos(θdθ + dθ + x lim t +(θ xt lim t + ( lim t + dθ xt + lim t (θ x ( x x ( rcsin + lim xt t ( ( t rcsin ( rcsin cos(θ cos(θdθ ( x xt rcsin + lim t x ( rcsin ( t (rcsin ( rcsin ( + (rcsin ( rcsin ( ( ( π + (π π rcsin (
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3 }t! t : () (f + g) f + g, (f g) f g (f g) f g + fg, ( f g ) f g fg g () [f(g(x))] f (g(x)) g (x) [f(g(h(x)))] f (g(h(x))) g (h(x)) h (x) (3) d vn n dv nv (4) dy dy, w v u x íªƒb N úb5} : () (e x ) e
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