<p>Definite Integrals Advanced Level Pure Mathematics Definite Integrals 1 Properties of Definite Integrals 3 Integration by Parts 4 Continuity and Differentiability of a Definite Integral 5 Improper Integrals 6</p><p>Definite Integrals</p><p>Definition Let f (x) be a continuous function defined on [a,b] divide the interval by the points</p><p> a  x0  x1  x2  ⋯  xn1  b from a to b into n subintervals. (not necessarily equal </p><p> width) such that when n   , the length of each subinterval will tend to zero.</p><p> th  [x , x ] i  1,2,, n lim(xi  xi1) f (i ) In the i subinterval choose i i1 i for . If n exists </p><p> and is independent of the particular choice of xi and  i , then we have</p><p> b n f (x) dx  lim (xi  xi1) f (i ) a n  i1</p><p> b  a Remark For equal width, i.e. divide [a, b] into n equal subintervals of length, i.e. h  , n</p><p> b n n b  a we have f (x) dx  lim f (i )h  lim f (i ) . a n  n  i1 i1 n</p><p>Choose  i  xi and xi  a  ih</p><p> b n b  a  f (x) dx  lim f (a  i) h a n  i1 n</p><p> b n1 b  a OR f (x) dx  lim f (a  i) h a n  i0 n</p><p> 2 2 2  1  2  1   2   n 1  Example Evaluate lim 1  1   1    1   n n   n   n   n  </p><p>Page 1 Definite Integrals Advanced Level Pure Mathematics</p><p> 1 1 1  * Example I = lim  ⋯  n n2  n n2  2n n2  2n2 </p><p>1 x x x *AL95II (a) Evaluate  a  b 1 , where a,b  0 . lim  x0 3 </p><p>(b) By considering a suitable definite integral, evaluate</p><p> 12 22 n2  lim  ⋯   3 3 3 3 3 3  n n 1 n  2 n  n </p><p> dx AL83II-1 Evaluate (a) ,  (x  a)(x  b)</p><p>  (b) 4 ln(1 tan x)dx , [ Hint: Put u   x .] 0 4</p><p>1   2  (n 1)   *(c) lim cos  cos ⋯ cos  n n  n n n </p><p>Properties of Definite Integrals</p><p>P1 The value of the definite integral of a given function is a real number, depending on its lower</p><p> and upper limits only, and is independent of the choice of the variable of integration, i.e.</p><p> b b b f (x)dx  f (y)dy  f (t)dt . a a a</p><p> b a P2 f (x)dx   f (x)dx a b</p><p> a b b P3 f (x)dx  f (x)dx  0 dx  0 a b a</p><p> b c b P4 Let a  c  b , then f (x)dx  f (x)dx  f (x)dx a a c</p><p>P5* Comparison of two integrals</p><p> b b If f (x)  g(x) x  (a, b) , then f (x)dx  g(x)dx a a</p><p>Example x 2  x , for all x  (0,1) ; </p><p>1 1 hence x 2 dx  x dx . 0 0</p><p>Page 2 Definite Integrals Advanced Level Pure Mathematics</p><p>3  2  2 Example Prove that (a) 2 x sin x dx    . 0 3  2 </p><p>  (b) 2 sinn xdx  2 sinn1 xdx 0 0</p><p>P6 Rules of Integration</p><p>If f (x), g(x) are continuous function on [a, b] then </p><p> b b (a) kf (x)dx  k f (x)dx for some constant k. a a</p><p> b b b (b)  f (x)  g(x)dx  f (x)dx  g(x)dx . a a a</p><p> a a P7* (a) f (x)dx  f (a  x)dx . a : any real constant. 0 0</p><p> a a (b) f (x) dx   f (x)  f (x) dx . a 0</p><p>2a a (c) f (x)dx   f (x)  f (2a  x) dx 0 0</p><p> b b (d) f (x)dx  f (a  b  x)dx a a</p><p>Exercise 7C</p><p> a a 1. * By proving that f (x)dx  f (a  x)dx (7c5) 0 0</p><p>  sin x evaluate (a) 2 (cos x  sin x)1997 dx (b) 2 dx 0 0 sin x  cos x</p><p> a a 2 (a) Show that f (x) dx   f (x)  f (x) dx (7c6) a 0</p><p>(b) Using (a), or otherwise, evaluate the following integrals:</p><p> 1 4 d 2 (i)  (iii) ln(x  1 x )dx  1 4 1 sin</p><p>Remind sin x,ex  e x are odd functions.</p><p>1 cos x, are even functions. x2 1</p><p>Page 3 Definite Integrals Advanced Level Pure Mathematics Graph of an odd function Graph of an even function</p><p>P8 (i) If f (x)  f (x) (Even Function)</p><p> a a then f (x) dx  2 f (x) dx a 0</p><p>(ii) If f (x)   f (x) (Odd Function)</p><p> a then f (x) dx  0 a</p><p>Definition Let S be a subset of R , and f (x) be a real-valued function defined on S . f (x) is called a </p><p>PERIODIC function if and only if there is a positive real number T such that f (x  T)  f (x) , </p><p> for x  S . The number T is called the PERIOD.</p><p>P9 If f (x) is periodic function, with period T i.e. f (x  T)  f (x)</p><p>  T (a) f (x)dx  f (x)dx   T</p><p>  T (b) f (x)dx  f (x)dx 0 T</p><p>T  T (c) f (x)dx  f (x)dx  0</p><p> nT T (d) f (x) dx  n f (x) dx for n  Z  0 0</p><p>Theorem Cauchy-Inequality for Integration</p><p>If f (x) , g(x) are continuous function on [a,b] , then</p><p> b 2  b 2  b 2   f (x)g(x)dx    f (x) dx g(x) dx  a   a  a </p><p>Theorem Triangle Inequality for Integration</p><p> b b f (x) dx  f (x) dx a a</p><p>Integration by Parts</p><p>Theorem Integration by Parts</p><p>Let u and v be two functions in x . If u'(x) and v'(x) are continuous on a,b, then</p><p>Page 4 Definite Integrals Advanced Level Pure Mathematics</p><p> b b b uv'dx  uv  vu'dx a a a</p><p> b b b or udv  uv  vdu a a a</p><p> sin kx * AL84II-1 (a) For any non-negative integer k, let uk  dx . 0 sin x</p><p>Express uk 2 in terms of uk . </p><p>Hence, or otherwise, evaluate uk .</p><p> (b) For any non-negative integers m and n , let I(m,n)  2 cosm  sin n dθ 0</p><p>(i) Show that if m  2 , then </p><p> m 1 I(m,n) =   I(m  2, n  2) .  n 1  (ii) Evaluate I(1,n) for n  0 .</p><p>(iii) Show that if n  2 , then </p><p> n 1 I(0,n) =   I(0,n  2) .  n  (iv) Evaluate I(6,4) .</p><p> AL94II-11 For any non-negative integer n , let 4 n In  tan x dx 0</p><p> n1 1   1   (a) (i) Show that    In    . n 1  4  n 1  4 </p><p>4x  [ Note: You may assume without proof that x  tan x  for x [0, ] . ]  4</p><p> lim In (ii) Using (i), or otherwise, evaluate n .</p><p>1 (iii) Show that I  I  for n  2,3,4,. n n2 n 1</p><p> n (1)k 1  (b) For n  1,2,3, , let an   . k 1 2k 1</p><p>(i) Using (a)(iii), or otherwise, express an in terms of I 2n .</p><p> lim an (ii) Evaluate n .</p><p>Page 5 Definite Integrals Advanced Level Pure Mathematics Continuity and Differentiability of a Definite Integral</p><p>Theorem Mean Value Theorem for Integral</p><p>If f (x) is continuous on [a,b] then there exists some c in [a,b] and </p><p> b f (x)dx  f (c)(b  a) a</p><p>Theorem Continuity of definite Integral</p><p> x If f (t) is continuous on a,band let A(x)  f (t)dt x [a,b] then A(x) is continuous at a</p><p> each point x in a,b .</p><p>Theorem * Fundamental Theorem of Calculus</p><p> x Let f (t) be continuous on [a,b] and F(x)  f (t)dt . Then F'(x)  f (x) , x  (a,b) a</p><p> g(x) Remark : H (x)  f (t)dt  H '(x)  f (g(x)) g'(x) a</p><p>Example Let f : R  R be a function which is twice-differentiable and with continuous second </p><p> x t derivative. Show that f (x)  f (0)  xf '(0)   f ''(s)dsdt , x  R . 0  0 </p><p> d x n * AL90II-5 (a) Evaluate f (t) dt , where f is continuous and n is a positive integer. dx 0</p><p>2 x 2 (b) If F(x)  et dt , find F '(1). x3</p><p> x  1 t 2 1  AL97II-5(b) Evaluate lim  3 e dt  2  . x0  x 0 x </p><p>AL98II-2 Let f : R  R be a continuous periodic function with period T .</p><p> d xT x (a) Evaluate  f (t)dt  f (t)dt dx  0 0 </p><p>Page 6 Definite Integrals Advanced Level Pure Mathematics</p><p> xT T (b) Using (a), or otherwise, show that f (t)dt  f (t)dt for all x . x 0</p><p>Improper Integrals</p><p> b Definition A definite integral f (x)dx is called IMPROPER INTEGRAL if the interval [a,b] of a</p><p> integration is infinite, or if f (x) is not defined or not bounded at one or more points in [a,b].</p><p> 1  1  x   1 Example e dx , x 2 , tan xdx , dx are improper integral. e dx 0 2 0  0 x</p><p> b Definition (a) f (x)dx is defined as lim f (x)dx a b a</p><p> b b (b) f (x)dx is defined as lim f (x)dx  a a</p><p> c  (c) f (x)dx is defined as f (x)dx  f (x)dx for any real number c .  - c</p><p> c ℓ ( Or lim f (x)dx  lim f (x)dx ) ℓ ℓ ℓc</p><p>(d) If f (x) is continuous except at a finite number of points, say a, x1, b where </p><p> b a  x1  b , then f (x)dx is defined to be a</p><p> c ℓ d ℓ lim f (x)dx  lim f (x)dx  lim f (x)dx  lim f (x)dx  ℓ  c  ℓ  d ℓa ℓx1 ℓx1 ℓb</p><p> for any c,d such that a  c  x1  d  b .</p><p>Definition The improper integral is said to be Convergent or Divergent according to the improper </p><p> integral exists or not.</p><p>Theorem Let f (x) and g(x) be two real-valued function continuous for x  a . If 0  f (x)  g(x)</p><p>  x  a then the fact that f (x)dx diverges implies g(x)dx diverges and the fact that a a</p><p>  g(x)dx converges implies that f (x)dx converges. a a</p><p>Page 7 Definite Integrals Advanced Level Pure Mathematics Example Determine whether the improper integrals are convergent or divergent.</p><p>0 a) xe x dx (p.109 7-45c) </p><p>2 dx b) 2 2 (p.112 7-50 b) 4 - x</p><p>1 dx c) (p.112 7-50 c) 1 x 2</p><p>Page 8</p>