10. cos (ax+b)sin(ax+b) dx = 12. sec x dx = tan x + c. 13. sec (ax+b)dx = tan (ax+b)+ c. 14. c sec x dx = - ctg x + c

Transkripsi

1 BAB XVI. INTEGRAL A. Integrl Tk Tentu. Rumus Integrl Fungsi Aljr. k k n = n +. ( + ) n = ( n + ). = ln + n + + ; n - n+ (+) + ; dn n -. ( f ( ) ± g( ) ) f ( ) ± g ( ) n. os (+)sin(+) = ( n + ) os n + (+) + ( + ) ( ). sin os sin +sin. se = tn +. se (+) = tn (+)+. se = - tg + 5. se (+) = - tg (+)+. Rumus Integrl Fungsi Trigonometri. sin = - os + 6. tn se = se + 7. tn se = -se +. os = sin +. tn = d os sin = os = - ln os + os. Rumus-rumus Integrl yng lin. = r sin ( ) + +. tg os sin = d sin sin = ln sin + ( = sin θ ; sin θ = ; θ = r sin ( ) ). + = ln sin( + ) = - os (+) +. = - ln + 6. os( + ) = sin (+) tn( + ) = - ln os(+) + 8. tg( + ) = ln sin(+) = r sin ( ) + = ln n 9. sin (+) os(+) = ( n + ) sin n + (+) + 6. = ln = ln

2 6. SOAL-SOAL INTEGRAL EBTANAS995. Hsil dri ( 8 + ) dlh A. 8 + B. + C. + D. 8 + E jw: ( ) = + = + Jwnny dlh B EBTANAS. Hsil 9 =. A. (9 ) 9 B. (9 ) 9 C. (9 ) 9 D. (9 ) 9 + (9 ) 9 9 E. (9 ) jw: Misl u = 9 - du = - du = 9 9 = - + = -. u + u = - (9 ) = - (9 ) 9 Jwnny dlh A UMPTN99. sin os =. A. sin. os D. sin B. os E. os - os C. sin Jw: r : n sin (+) os(+) = ( n + ) sin os = ( + ) sin + Cr : Misl: u = sin du = os = sin + sin os (sin ) os u du = u + = sin + sin n + (+) + u. du = - u du Jwnny dlh C -

3 UAN. Hsil sin( + ) = A. os ( + ) D. os ( + ) B. os ( + ) E. - os ( + ) C. os ( + ) jw: u = + du = sin( + ) = sin u du (kren du = ) = - os u + = - os ( + ) + Jwnny dlh C UAN sin 5. = EBTANAS 6. Hsil os. sin 5 =. A. - os7 + os + 6 B. - os7 os + 6 C. os7 os + 6 D. os7 + os + E. os7 os + Jw : sin A os B = sin (A+B) + sin (A-B) sin A os B = sin (A+B) + sin (A-B) os. sin 5 = sin 5 os = { sin (5 + ) + sin (5 ) } = ( sin 7 + sin ) A. sin + C. sin + E. os + os. sin 5 B. os + D. os + Jw: sin 7 + sin = -. 7 os os + Misl ; u = = = -. os 7-6 os + du = - = - sin = - = os u + sin u du Jwnny dlh B UN6 7. Nili dri = = os + A. B. 8 C. 6 D. 5 E. Jwnny dlh D -

4 Jw: + = (+) EBTANAS99 π 9. sin(-π ) = + + = ( +) = + A. - B. - C. D. E Jw: ( + ) + π sin(-π ) = =. = 8 Jwnny dlh B UAN7 8. Dikethui ( + + ) = 5, nili = A. - B. - C. - D. E. Jw: = - os (-π ) = - os (π - π ) (- os( - π ) ) = - os (- os - π ) = = - - = - π ( + + ) = + + * os =, * os - π = os(π π ) = - os π = - os 6 = - ) = ( + + ) = 9 - ( + + ) = 5 ( + + ) = Kit lkukn uji o nili (tril & error) : Msukkn nili + + = tidk memenuhi + + = = memenuhi = = -6 tidk memenuhi mk =, sehingg = Jwnny dlh D jwnny dlh A UN6. Lus derh tertutup yng ditsi oleh kurv y = - dn gris y = dlh. A. 6 5 stun lus D. 6 5 stun lus B. 6 5 stun lus E. 6 5 stun lus C. 6 stun lus Jw: y = -.() y = y = +.() -

5 sustitusi () dn () : + = = ( - ) ( + ) = titik potong di = (ts ts) dn = - (ts wh) UN7. Lus derh yng dirsir pd gmr di wh dlh.. stun lus : skets gmr untuk meliht posisi kurv dn gris, pd gmr terliht posisi di ts dlh gris, sehingg untuk menghitung lusny dlh persmn gris dikurngi kurv (kondisi selikny pil kurv di ts gris) A. 6 5 C. 7 E B. D. 6 6 jw: Titik potong kurv dn gris : (y y) ( + ) ( - ) ( ) ( ) = = - (7 ( 8)) + (9 ) + 6(-(-)) = - (5) + (5) + 6(5) = = = 6 = 6 5 stun lus Jwnny dlh D = = ( + ) ( ) = Titik potongny dlh = - (ts wh) dn = ( ts ts) lusny = (pers.kurv pers gris) = 6 - (9- ) ( +) (9 - - ) (6 - - ) - -

6 = 6 (-(-) ) - (8 ( 7)) - ( 9) = (5) - ( 5) = - + = 6 = 5 5 = 6 6 Jwnny dlh A UAN. Derh yng ditsi oleh kurv y = dn gris + y = diputr mengelilingi sumu sejuh 6. Volume end putr yng terjdi dlh A. 5 π stun volume B. 5 5 π stun volume C. 5 π stun volume D. 5 π stun volume E. 5 π stun volume Jw: sustitusi () dn () = + = ( + ) ( ) = = - (ts wh) tu = (ts ts) (liht pd gmr) Menri volume : V = π = π = π = π = π (y - y ) { (-) - ( ) } (( - + ) - } ( + - ) ( ) 5 = π ( ) = π {(- ( ( )) + ( (-(-8))-(-)+(-(-))} 5 = π {( (-) +. ) = π ( ) +5 = π (- + ) = π = π = π stun volume 5 5 jwnny dlh D Menri titik potong: y = () + y = y =..() - 5

7 UN7. Volume end putr il derh yng ditsi kurv y = - + dn y=- + diputr 6 mengelilingi sumu y dlh. A. 8π stun volume B. π stun volume C. π stun volume 8 D. π stun volume 5 E. π stun volume Jw: y (y - ) = didpt y = tu y = ( terliht pd gmr) menri Volume: kren diputr terhdp sumu y rumusny menjdi: V = π ( - ) dy = π = π = π = π ( y) { (-y) (6 8y + -y y } dy (6 y 6 + 8y (y - y ) dy ) } dy y ) } dy π = ( y - y ) π 6 π 96 6 = ( - ) = ( ) π 8 = = π stun volume Menri titik potong: Persmn kurv y= - + = y () persmn gris y = - + = y y =..() sustitusi () dn () ( y) = y = y (-y) = 6 y 6 8y + y = 6 y y+ y+ y = - y + y = y - y =

8 8. +. Integrl Prsil = r tn +. Jik f() > (Kurv di ts sumu ) u dv = uv - v du Didpt dri : y = u.v dimn u = g() dn v = h() y = u v + u v = v u + u v dy du dv = v. + u. dy = v du + u dv d (u.v) = v du + u dv (diklikn ) L f ( ). Jik f() < (Kurv di wh sumu ) ( u. v ) d v du + u dv u.v v du + u dv u dv = uv - v du B. Integrl Tertentu f ( ) = F() = F() F() L = - f ) ( f ( ). Jik f() > dn f() < (Kurv segin erd di wh sumu dn segin linny erd di ts sumu ). Lus Derh Antr Kurv dn Sumu- sumu Koordint Lus derh yng ditsi oleh kurv y = f(), sumu dn gris-gris = dn = sert =g(y), sumu y dn gris-gris y = dn y = dpt diedkn s L = - f ) ( + f ) ( + f ( ) f ( ) -

9 d. jik g(y) > (kurv erd di seelh knn sumu y) i L g ( y) dy e. jik g(y) < (kurv erd di seelh kiri sumu y) L = - g ( y) dy + g ( y) dy g ( y) dy + g ( y) dy. Lus Derh Antr Du Kurv. Di ts sumu L = - g ( y) dy g ( y) dy f. jik g(y) < dn g(y) > (kurv segin erd di seelh kiri sumu y dn segin linny erd seelh knn sumu y) L y - y ( y y) -

10 . Di wh sumu L = - y - { - ( y y) y }. Di seelh knn sumu y y - y L dy -. Volume Bend Putr dy ( ) dy. Diputr terhdp sumu mk, V= π y. Diputr terhdp sumu y mk, V= π dy -