ΗΛΙΑΣΚΟΣ ΦΡΟΝΤΙΣΤΗΡΙΑ. Θετικής - Τεχνολογικής Κατεύθυνσης Μαθηματικά Γ Λυκείου Ολοκληρώματα ΥΠΗΡΕΣΙΕΣ ΠΑΙΔΕΙΑΣ ΥΨΗΛΟΥ ΕΠΙΠΕΔΟΥ ΣΤΕΦΑΝΟΣ ΗΛΙΑΣΚΟΣ

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Καμβύσης Ηλιόπουλος
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1 ΗΛΙΑΣΚΟΣ ΦΡΟΝΤΙΣΤΗΡΙΑ ΥΠΗΡΕΣΙΕΣ ΠΑΙΔΕΙΑΣ ΥΨΗΛΟΥ ΕΠΙΠΕΔΟΥ Θετικής - Τεχνολογικής Κατεύθυνσης Μαθηματικά Γ Λυκείου Ολοκληρώματα ΣΤΕΦΑΝΟΣ ΗΛΙΑΣΚΟΣ e-mil: info@iliskos.gr

2 Fl] = f]! D G] = F] + c! R G] = F] + c! R f] d f] d = F] + c! R fl] d = f] g+ c! R d = c d = + c f d = + c f fl ] g ] g ] d = + c + + f d =,!- + f fl ] g ] g ] d = + c,!- + d ln c, fl = + > ] d = ln f ] g + c f]

3 f] f] ed = e + c e fl] d = e + c d f] f] = + c fl] d = + c ln ln hnd =- vyo+ c ^ hnf] hfl] d =- vyo^f] h + c vyod = hn+ c vyo^ f] hfl] d = hn^f] h + c =- v{ + c hn = f{ + c vyo ] + ] = + ] ] mf] d = m f] d, m! R ^ f g hd f] d g] d fl] d =- v{ ^f] h + c hn ^f] h fl] d = f{ ^f] h + c vyo ^f] h ^ mf + ng hd = m f] d + n g] d, mn!, R

$R ^ f g hd f] d g] d fl] d =- v{ ^f] h + c hn ^f] h fl] d = f{ ^f] h + c vyo ^f] h$

4 d d = = + c fl] f d f - = f l ] g ] g ] g d = + c f ] - + = ln d = ln + c f l] g d = ln f ] g + c f] d v n v n v d = = + n + c > n + v v f n v f fl d = f n fl + ] g ] g ] g ] g ] d = v + c, f] > n + n v >, f] > f] hn vyo vyo hn = - vyo vyo = +

5 fl] g] d = f] g] - f] gl ] d 4 4 e d e l ] - g = l ] - gd = 4 4 e e d 4 4 ] - g - 4 ] - gl = ] g ] g ^ hld = ] gl vyo ^ h I = e hnd = ] e gl hnd = e hn- e hn e hn- e vyod = e hn- e d = e hn- e vyo+ e - hn d = e hn-e vyo- I + e hn- e vyo I = e hn- e vyo + I = + c

6 ln ln ln d = l ln d = ln - d = ln ln ln - d = ln - + c ln] + gd = ] + gl ln] + gd = ] + g ] + g- ] + g d = + ] + g ] + g- d = ] + gln] + g- + c P] Q] P] d Q] U] P ] d + d P] g Q] U] Q] U] d Q] U] Q ] g Q] Q] Q] ^ - th A ^ - th - 3 A A = + ] - g] - g - - Q] - t

7 l ^ - th l ^ - th A A A - t + f i t i - t l l ^ h ^ ih A, A, f, A l - 3 A A A3 = + + ] - g ] - g - ] - g - Q] ^ + + chd = -4c < ^ + + ch A i + Bi + + c A, B - A A + B = + ] - g] - + g Q] ^ + + ch l D = -4c < ^ + + ch l A + B A + B A l + Bl + + f + l + + c ^ + + ch ^ + + ch A, B, A, B, f, A l, B l - 5 A A + B A 3 + B3 = + + ] - g] + + g ] + + g U] Um] Q ] / g Qm] U] Um] = /! R Q]! Q] Qm]! RQ]!! R Q]!

8 U] d Q] I = - t d I = d ln - t = - t + c 3 I = - d ] - g] - g - 3 A A = + ] - g] - g = A] - g+ A] - g = - = A+ A] g + A =- = - = A] - g + A + A = = + ] -g] -g - - I = - 3 d = ] -g] -g - + d l I = ln - - ln - + c I = I = l d ^ - t h l H - l+ l ^ - tih l d = ^ - tih d = + c ^ - tih - l + I = - 3 d ] -g ] -g - 3 A A A3 = + + ] - g ] - g - ] - g = A] -g] - g+ A] - g+ A3] - g = - = A+ A] - g + A3 + A = = - = A+ A + A3] g + A3 =- - 3 = A] -g] - g+ ] - g-] - g +

9 - 3 = ] A - g + ] 4-3Ag+ A -5 A - = + A = = + + ] - g ] - g - ] - g - I = - 3 d = c + - md ] -g ] -g - ] - g - + I = ln ln c I = A + B dd = -4c < + + c A + B = t^ + h = t^ + + chl A + B t^ + + chl I = d = d = + + c + + c t ln + + c + c I = -- d ] - g] + g A A B = + ] g] g = A] + g + ] A + Bg] -g = - = A] g + ] A+ Bg + A =- -- =- - + ] A+ B g] -g + -- = ] A - g + ] B - A g- -B A - = A ) + ) -- B =- B = =

10 = ] -- - = + ] - g] + g I = d = + d + ] - g ] + g - + l I =-ln - - ln] + g + c A B I = + d ^ + + ch D = - 4 c < l l H A + B = t^ + h= t^ + + chl A + B t^ + + chl I = l d = l d = ^ + + ch ^ + + ch -l t^ + + chl^ + + ch d = ^ + + ch t - l + - l+ + c I = f^ g] tghgl ] tgdt u = g] tg du = gl] tgdt I = f] ugdu + I = F] ug + c + I = F^g] tgh + c I = f] d = g] tg & d = gl ] tgdt I = f^ g] tghgl ] tgdt g t g

11 t = h] h] = h ] tg & d = ^h ] tgh dt - - I = f^ h ] tgh^ h ] tghl dt - - l t = h] vyo hn = -, vyo = + vyo I = hn d I = hn d = - vyo hn d = - + c 4 hn + vyo = hnu = vyo & du =-hnd vyou = hn & du = vyod I = hn d I = hn 3 d = hnhn d u = vyo & du =-hnd u = vyo + u = vyo + u = - hn + hn = -u 3 u I = -] - u gdu =- u + + c + 3 vyo I =- vyo+ + c 3 I = hn 3 vyo d I = hn 3 vyo d = hnhn vyo d u = vyo & du =-hnd u = vyo + u = vyo + u = - hn + hn = -u

= hnhn d u = vyo & du =-hnd u = vyo + u = vyo + u = - hn + hn = -u 3 u I = -] - u gdu =- u + + c + 3 vyo I =-

12 3 5 4 u u I = -] - u gudu = ]- u + u gdu =- + + c vyo vyo I =- + + c d I = + d u u = + + = + + = u - & d = udu 5 3 u u I = ] u - guudu = ] u 4 - u gdu = - + c ^ I = + h ^ h c u = e + = ln u & d = u du e - I = d e + u = e + = ln u & d = u du I = u - u + u du u - A B u = u + u + ] g u + u- = A] u + g + Bu = - = A+ B + A =- =-- = A+ B] - g + B = u - u = - u + u + ] g u + I = u - u du = - u u + du = ] + g u + l - ln u + ln u+ + c + I =- ln e + ln e + + c ln u ln e = + = & d = edu

13 I = ] ln + ln -3gd u ln e = + = & d = edu I = ] u + u- 3gedu u = u u 3 e u u ] + - g - ] u + gedu = u u u ] u + u-3ge -] u + g e + edu = ] u + u-3ge u -] u + ge u + e u + c + ln ln ln I = ] ln + ln -3ge -] ln + ge + e + c - = hnu & d = vyou - u u d hn vyo = & =- hn u + = f{ u & d = u du vyo / v l l = v = / f^plhdd = - v p p fp v v lim/ f^p hd f] d v " + 3 l l = l =

14 I = f] d f] = g]! f] d = g] d = f]! ^, h f] d = 6 = F^ h - F] g mf] d = m f] d, m! R ^ f] + g] hd = f] d + g] d ^ mf] + l g] hd = m f] d + l g] d, lm!, R f ] g d = f] d =- f] d

^, h f] d = 6 F] @ = F^ h - F] g 6 @ mf] d = m f] d, m!

15 c c f] d + f] d = f] d H! f] d H G! f] d G f] d > f] d < f] H f] =! f] d > f] G g]! f] d G g] d f] G g] + f] - g] G & ^f] - g] hd G + f] d - g] d G + f] d G g] d f] G g] f] G g] f] d G g] d

16 f] d H g] d f] gl ] d = 6 f] - fl] g] d 6 f] = f^hg^ h-f] gg] g f] d = g] tg & d = gl ] tgdt = g] g, = g^h f] d = f^ g] tghgl ] tgdt

17 I = f] d - - I = f] d = f] d + f] d u =- & du =-d = & u = =- & u = I = -f]- ugdu + f] d = f] - ugdu + f] d f] - = f] I = f] ugdu + f] d = f] d I = f] d - - I = f] d = f] d + f] d u =- & du =-d = & u = =- & u = I = -f]- ugdu + f] d = f] - ugdu + f] d f] - = f] I = - f] ugdu + f] d = I, I f^hnh f^hnh u = π - & du =-d f] vyo f] vyo u = π - & du =-d f^hnh f] vyo π u = - & du =-d f] f] l - u = l - & du =-d

u = π - & du =-d f] vyo f] vyo u = π - & du =-d f^hnh f] vyo π u = - & du =-d f] f] l - u = l - & du

18 :, ϖ D π π f^ hnhd = f] vyod π u = - & du =-d π = & u = π = & u = π π f^hnhd = -fchn - uldum = π π f] vyougdu f - ] f] u = f ] + = f] ug & d = f ] ugdu - l - f ] d = f] d - f + f - ] g = ^h = f^ h = + f ] g = u = f ] + = f] ug & d = f ] ugdu - l - = & u = f ] g = - = & u = f ^h= - l f ] d = uf ] ugdu = 6 uf] ug@ - f] ugdu = f] g - f^ h + f] ugdu = - + f] ugdu = f] ugdu - f ] d = f] d

19 I, I, f, I v, f v - v Iv = ed! N v- N v v v- Iv = ed = 6 e@ - v ed + Iv = e- vi I3 = e-3i I = e-i I = ed = - ed = e- = I3 = e-3] e- Ig =- e+ 6I = 6- e f] tgdt! F: D " R F] = f] tgdt! D Fl] = f]! D F] = f] tgdt f d l ] g l =-f]! D ] lg l +

= e-i I = ed = 6e @ - ed = e- 6e @ = I3 = e-3] e- Ig =- e+ 6I

20 ] l+ g F] = f] tgdt F] f] f] f] = t -9dt h] tg = t -9 A = ^-3, - 3@, 63, + 3h H A = 63, + 3h -3 < < 3 G-3 A = ^ -3, -3@ f] tgdt h] f] = g] tgdtfl] = g^ h] hhl ] f = { ] g ] g g] tgdt h] f] = g] tgdt + g] tgdt & {] h] fl] =- g^ h] hhl ] + g^ {] h{ l] f] = h] g] tgdt h] f] = h] g] tgdt & fl] = hl] g] tgdt + h] g] gft ^ ^, hh

$f = { ] g ] g g] tgdt h] f] = g] tgdt + g] tgdt & {] h] fl] =- g^ h] hhl ] + g^$

21 f] = g] tgdt u u = t + t = & dt =- du t = & u = t = & u = g] ug f] = - du =- g] ugdu & ' ' =, = E = f] d ' =, = f] = y = '

22 = = 4 f] = f E = ] gd - ] gd + ] gd E = + + = =, = E = f] - g] d =,= h] f] g] -= g f] g] - = f] = M^, -5h 'y y f ] M^5, hy- ]- 5g = fl] g] - g y =- 6+ 7

23 h] = f] - y = ] g = h] =-] - g G G G E =- h] d =-6 -] - = 8 3

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