Calculus I- Module 9 - Indefinite Integrals

Module a A function F indefinite 9.1 is an antiderivative of f x y If Interguand Integral Sign Frg K f x integration constant Antiderivative Snax constant Multiple Combo integrable Functions s a P Constant Rule Skax Linear If h F X if Variable of integration BASIC RULES Power Rule integrals II Kx B veal C th ne Kf xldx rule kffcxldx afcxlt Bgcx13dx afflxldxt.BGgadx Rule Evaluate A f5dx Sx B fadt Tf t C C x'dx D fusdu fi fi Ef3xdx 3 Xdx Constant Rule C Constant Rule te e te te 3 C Ex 53 C Wit C C Power Rule Power Rule Constant multiple P Power Integrating polynomial functions A f 2x 5 dx 2SXdx fSdx 2 C Sx Power b constant XtSx B Bx x Tldx Simplify fxdx GTdx 3 x'dx C 3x 5 3 4 gents Txt C x dx x 3 D C Tix 3 x Linear combo dx 5G4dx 4 s 1 urfk.is iFax 27 uy 3 4 3 4 25.3 15 3 tf x C ax Rewrite and Simplify before integrating Original Rewrite l S dy X 2 S Adx xtdx g g g ZE'dt Simplify integrate Ex AS 2x 1 x2 2 Ite It dx 2 E 43 4 Ete te 2 3 x2 4 dx te 2 x dx 2fx3dx fx2dxt4fxdx f2dx 2É Y B f x Idx dx f 1 4 dx Sx dx 3 x dx It C.SE xdx fdx f2fdx sfsx Is gx dx 2 dx f 2x x 2x'idx te SCHE 2 E x IOT F x't C C dx 3 Integration rules for trig functions Sin xdx f Sinaxdx cosx te f Cosxdx Sinx te f see'xdx tanx C I Secxtanxdx See xtc f est xox cot xtc tacos ax C dtanaxt ii.is fsectaxdx fcseaxdxi dcotaxt fcsexcotxdxs.es fescaxcotaxdx descaxte ex A C Ssinxt cosxidx 5 Sinxax 5C cos X 3 Secttan tide flosxdx 7 Sin Xt e ScosX 7 B 7 3ft at Sin x C Sect tant Sect C 3 t3_Sect C 35in2x Cos E dx 3 Sin2xdx Sos 3 Cos2 ECos2x D 20 sin'xdx 3 S Ex 3 sin E dx E C 3sin SEE 3 lax 2 Sesexcotxdx S 3dx 1 2 COEX 3 2cotX 3 integration rules for exponential t f t dy en ly te A f Zet 4 dx 2 Je'dx e te 2 exdx 4 f e'd x C Je f Edt de IdITgsinsa I sina.it I 2 43 e c x lx 4 L tcos3x Iscos 3x t xt4cos3x E ta e 4 2ex c seem a function f x'd x 2 ex 4 B f Ce't E de logarithmic 104 8 de EE t 10th 8 In El te ILE 21 8 In El te C C te Solve initial value Problems Differential equations Differential equation dat fix or FX f Initial Condition yea b or Fca x b A Diff equation with an initial condition is called an initial problem IVP the Solution is Solve Ex 3 2 4 13 2 E 4 43 C FIX y fixidxte 2x W initial Condition 4107 2x dx 2 E te Xtc 4101 1013 COP C f O C I Y x3_x Solve F t 1 Z E i y 112 E dx 2 E It E Ite y ZE Etc 4117 2 1 It C O 3 0 3 Y E_ 2E I valve Motions problem The Velocity Vitt is known tofind position S E V14 alt V E So So is initial position SCO Acceleration anti is known sett wi some IVP to find velocityv ite we solve IUD V01 Vo Vo is initial Velocity A Stone is thrown upward frominitial height 100ft wi initial velocity of 48ft S Find height as function of time t Let t O as Starting initial time V 01 48 5101 100 t V VCE VCO Acceleration 32 f 32dL 32 48 Alt 32ft s te 32101 2 48 0 48 V _32tt C 4g Given the following Velocity funition of an object moving along a line find the position fruition with gr en v its Est 4 5107 2 510 te se 2 ft FIX Lit C 4 lol te e s sh je.tt g the accelerationof a car is given by the gravity which is 9.8 mis Astone is thrown vertically from initial velocity Gomis from a cliff 100 Above a river A Find velocity 20 vets of object B find position Sct C find Max height reached A Let EO 60 VCO 5101 100 a E C Cause Free Falling 9.8 mis f Vitt 9.8dL V O 60 3 60 60 O 9.82 t C 9.8 O VLtk 9 60 C B S E 9.8 t V E I C A.IE Stel S161 Y 9.8 S 6 1 60 dt SO 4.922 60 E te 4 a 1012 60 o 60 0 100 0 100 6 6.1 4.9 6.112 6016.1 Maxheight e 2 100 tt6ottou K SE t 60 60 Etc 9.8 S 101 100 C C 100 283 IM 283 7m GI Use Sigma summation notation to Sigma write b evaluate summation is used to express sums in compact n on terms a an is written Az way the Sum or 7 an a ta ta t an is the index of Summation an is the 4th term of the sum b the upper b lower bounds of summation are and I respectively Where he Examples I 2 4 Ifk Éi 1 2 3 4 21 32 42 52 FIX AX FIX AX 6 5 7 6 FIX JAX FIX AXt FAN AX Properties e constant I Constant multiple Sum Diffrence Rule Theorem 1 É an E art by be Solution formulas É 1 Sum of Constant c 2 Sum of first 3 Sum of CE an can n É integers on c It k Squares of first n integer 4 Sum of Cubes of first n integer IL If k at hint It k3 Evaluate a 5k 1 k 5 S S 251225 EI I 1 257 137 1600_ Sum Difference 1 25 25 12 b I rule b If I fi Citi too IF it too too 1001200 51 5 Factor out too Sum Diffrence tension z i Approximate Area under A X IO on interval a Goal is graph of f b X axis interval Yo X I o length If a b Xi X If AX of rectangle on Summation notation Can i Xn Exa X of Subinterval the Area Ik b of region R bounded b X a to X b We begin into n sub interval of lengt from A IK AX In curve to approximate A by deviding the a by kth Subinterval where K height A a is ÉF 1,2 is h f Iu Iu AX Sum is Riemann Sum the left endpoint Interval Yes Xu the above sum is called the left Riemann Sum Xk 1 Ia Xu the rightendpoint interval Xr i Xu the above sum is called the Right Riemann Sum Approximating Area under curve using Riemann Sums Estimat Area A under thegraph of F on interior i 20,2 4 Using left P Right Riemann Sums With n 910 s14 911313 81 lr.ES the length of subinterval AX 2 BI Of The Left Riemann Sum If F Xk Ax f Xo t FIX t f 1O f 1.5 DX 0.5 14310.5 o t f 10.5 f 3 4.5 sq units LI FAN DAX Ffa 5.5 Right 1 FIXa AX Lf x If 10.5 3 F X2 I F 1O 4.5 5.5 f 6.0 19710.5 IS sq FX on its 15 0.5 t Fan B FROLD AX