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INDEFINITE INTEGRAL DEFINITION lnlegralion i [he reverse process ol cillereantialai. If Pes =f fx) then Fixd is antideryvative af the funetor fix’. 2. lf Fix) is an antidervative of fic thon Pie) + co whore c is a constant. is also antderivative af fix) thus function posses infinitely many antiderivatives. Symibolically we can represent it a8 ffi dx = Fixi+c. Hence we nave i FIX} c| Fix| jtixi dx Fix) )c where cis the constant of integration. Rules of Indatinite Intagration: i [KifG de k( fix) dx whera k is a conatant, i) (fi pot hte i ba}de ioe fii) dx fiat da. Loo de. wes : : Ite ik iit IF ffi dx = Fi[xl +. ten [Flax by dx a“ ‘ eo Vas Gf. Example i: Evaluate ii} [tan® dat. (ii) Je" [ras - sinvide. (ity Feet + eFax. jv (sin dacos2xce . Solutions: ii} [tan’xdx [sac’x (jdx =tanx-x+c. iii fe*tonsx - sinxhex, Here e‘icosx - gin is te derivative of e” cosx, = |= a’ o0s« 14. ith ae! + xd, Here ae 2") is the derivative of a gee —l=(1-x" Fac. frei [si ax aos 2xcds . When solving such problems it is expedient to use tre following triganometric identities : sin(rrix] cus(rie| = (12) [sintrn - je + Sint + nis] sin(rrie] sire) = (12) fomestrn - ape - crstin + ris] cosime) cosinal = (172) [cosim - nix +sos{m + nix] Here sin ax cos 24 = (125 [si isl + sit x| 1 |e Bx 4 . . — |. 3 J lsinSx = sinx }ox = ~ +o08x| +5, BASIC FORMULAE Antidervatives or integrals af some of the widely used functions (integrands) are given below. dit 3 : =. “nt . x = |x dx le.ne- dx. nid) . nid . 4 -1 - Fx) — ee ee da x “x - 4 aja ja dasa" +c cx 
