Page 1 : Evaluation of Definite Integrals by, Substitution, , When the variable in a definite integral is changed due to, substitution, then the limits of the integral will accordingly be, changed., For example, to evaluate definite integral of the form, , J2flel0] -g (ode, we use the following steps, , Step1 Substitute g(x)=t so that g’ (x) dx =dt, , Step II Find the limits of integration in new system of, variable, Here, the lower limit is g(a), the upper limit, , is g(b) and the integral is now Sic fe (t) dt., , , , Steplll_ Evaluate the integral, so obtained by usual method., , Properties of Definite Integrals, 0 | reyae=0, , ti) J fea dx= JT peoat, tai) fla. dx =— J "709 de, , », , iv) J foaax =f" flajdic+ f° flgdx ++ g fod dx, where, 0 <c, <6, <..., ©) @) JP fod de= f? fa +p- 9) dx, ) ff face JP flax) dx, (wi) f° foadx, , f° foadwif fi) = fe9, = ie. f(x) is an even function, jo, itf-y=—F00), , i.e. f(x) is an odd function, , <e, <B, , , , (vii) f food -j flax | flea x)dx, , , , 2 f° foe if flea. — x, , (wilt) f°" so0 de = if fla — x) =—f(x), , , , tif seydx=B-«) “fl@-0) x+ apex, (x) If f(xisa periodic function with period T, then, @ ["foode=n Jp@jdxner, ) JP Foy dx=(B-0) Jfodds, aber, ©) [2 Fe) d= J? fd der, , a sar, , (xi) Some important integrals, which can be obtained with, the help of above properties., , 3) flog sin xdx = ‘tee cos xdx = Eog(2 ja, , yt log(1 + tan x)dx = Flog 2, , (xii) If a function f(x) is discontinuous at points x,,%).....%,, in (a,b), then we can define sub-intervals, (4, X,)s (4% Jeeer(% p15 Xp) (%,.b) such that f(x) is, continuous in each of these sub-intervals and, , J peas = f fiside + f foes tf flogdx + j fod., , , , Leibnitz Theorem, If function 6(x)and y(x) are defined on [a,,B] and differentiable, on[a,B]and (0) is continuous on [y(a), 6(B)}, then, , A ipeeioar] {= wos} revoor | «oo } foe, , ate), , Walli’s Formula 7, , This is a special type of integral formula whose limits from 0, to /2 and integral is either integral power of cos x or sin x or, cos xsin x., , JP sine dx = [°" cos" x dx, , , , [(n-1) (n-3)(n-5) ... 5. 31m if n=2m (even), n(n —2)(n- 4)...6-4-2 2, (n-1)(n-3)(n- 6-4-2, , He-6- bt , ifn=2m +1 (odd), , where, nis positive integer., Jin” x-cos" xd, (m-1)(m -3)...@or1).(n-1)(n-3)..., (m + n)(m+n—2)...(20r1) 2°, when both m and n are even positive integers, =} (m-1)(m-3)...(20r1)-(n—1)(n—3)...(20r1), (m+ n)(m + n—2)...2or1), , when either m or n or both are odd, , Qort) x, , , , positive integers, , Inequalities in Definite Integrals, (i) If f(s) > g(x) on [a.,B}, then fi ‘fo dx > Jiao dx, (ii) If f(s) 20 in the interval [a.,B}, then i fod dx 20, , (iii) If f(), g(x) and h(x) are continuous on [a,b] such that, , 80) < f(x) < HO, then fg(x)de < f!f(aide < Abad