Solution - Career Point

If I n =∫x n . e ax dx, then I n =nx eaIf I n =∫(log x) dx , then I n = x log x – xIf I n =∫1dx, thenlog xI n = log(logx) + logx +2(log x)2.(2!)ax+– anIn–13(log x)3.(3!)nIf I n =∫(log x) dx ; then I n = x(logx) n – n.I n–1+ ...Successive integration by parts can be performedwhen one of the functions is x n (n is positive integer)which will be successively differentiated and theother is either of the following sin ax, cos ax, e –ax , (x+a) m which will be successively integrated.Chain rule :∫u .v dx = uv 1 – u´v 2 + u"v 3 – u"'v 4 + ....+ (–1) n – 1 u n–1 v n + (–1) n ∫u n .v dxwhere u n stands for n th differential coefficient of uand v n stands for n th integral of v.∫axxe sin(bx + c)dx =cos(bx + c)] –(acos (bx + c)] + k∫ax2eax+ bxe cos(bx + c)dx =sin(bx + c)] –(asin (bx + c)] + k∫xeaxa=(loga)∫xeax2eax+ bsin(bx + c)dxxa=(loga)2+ b2cos(bx + c)dxx2+ ba cos x + bsin x∫dxccos x + d sin x222))22aaxe2ax+ b2[a sin(bx + c) – b[(a 2 – b 2 )sin (bx + c) – 2abx.e2ax+ b2[a cos(bx + c) – b[(a 2 – b 2 )cos (bx + c) – 2ab[(loga)sin(bx + c) – b cos(bx + c)] + k[(loga)cos(bx + c) + b sin(bx + c)] + kac + bd ad − bc= x + log |c cos x + d sinx| + k.2 2 2 2c + d c + dsinReduction formulae for I (n,m) =∫cosn−1nmxxdx is1I (n,m) =m − 1. sin x ( n −1)– .Im−1(n–2, m – 2 )cos x (m –1)Definite Integral and Area Under Curves :1 bThe number f(c) =− ∫f (x)dx is called the(b a) amean value of the function f(x) on the interval [a, b].If m and M are the smallest and greatest values of afunction f(x) on an interval [a, b], then m(b – a) ≤∫ baf (x)dx ≤ M(b – a).If f 2 (x) and g 2 (x) are integrable on [a, b], then∫ ba⎛f (x)g(x)dx ≤ ⎜⎝∫abf2⎞(x)dx ⎟⎠1/ 2⎛∫⎜⎝ab1/ 22 ⎞g (x)dx ⎟⎠Change of variables : If the function f(x) iscontinuous on [a, b] and the function x = φ(t) iscontinuously differentiable on the interval [t 1 , t 2 ] anda = φ(t 1 ), b = φ(t 2 ), then∫ bat2t1f (x)dx =∫f ( φ(t)φ´(t)dtLet a function f(x, α) be continuous for a ≤ x ≤ b andc ≤ α ≤ d. Then for any α ∈[c, d], if I(α) =∫abf (x, α)dx, then I´(α) =∫f ´(x, α)dx, whereI´(α) is the derivative of I(α) w.r.t. α and f´(x, α) isthe derivative of f(x, α) w.r.t α, keeping x constant.∫ ba∫bf ´(x)dx = (b – a)∫f [(b − a)t + a]dtf (x)dxf (x) + f (a + b − xa )10ab= 21 (b – a)The area of the region bounded by y 2 = 4ax, x 2 = 4by16abis sq. unit.3The area of the region bounded by y 2 = 4ax and y =28amx is sq. unit.33mThe area of the region bounded by y 2 = 4ax and its8a2latus-rectum is sq. unit.3The area of the region bounded by one arch of sin(ax)or cos (ax) and x-axis is 2/sq. unit.Area of the ellipse (x 2 /a 2 ) + (y 2 /b 2 ) = 1 is πab sq. unit.Area of region bounded by the curve y = sin x, x-axisand the line x = 0 and x = 2π is 4 sq. unit.XtraEdge for IIT-JEE 47 APRIL 2010