Page 5 - CH-7-SLRC-HOME ASSIGNMENT

B. DEIFINITE INTEGRALS
01. Integral as a limit of a sum

b 1 b a
 f(x)dx (b a) lim [f(a) f(a h) f(a 2h) .... f(a (n 1)h)], where h










a n n n
Another form
b
 f(x)dx  lim h [f(a) f(a h) f(a 2h) .... f(a (n 1)h)], where nh = b – a.








a h 0
02. Properties of Definite integrals
b a
(i)  f(x)dx    f(x)dx
a b
b b
(ii)  f(x)dx   f(y)dy
a a
b c b
 
(iii)  f(x)dx   f(x)dx   f(x)dx, for a c b
a a c
b b
(iv)  f(x)dx   f(a b x)dx
 
a a
a a
(v)  f(x)dx   f(a x)dx

0 0
2a a a
(vi)  f(x)dx   f(x)dx   f(2a x)dx

0 0 0
2a a

(vii)  f(x)dx 2 f(x)dx,if f(2a x) f(x)



0 0
= 0 if f(2a x)  f(x)

a a



(viii) f(x)dx 2 f(x)dx if f(x) is an even function
 a 0
= 0, if f(x) is an odd function.
b b b

(ix)  [f(x) g(x)]dx   f(x)dx   g(x)dx .
a a a
03. Some important results which are often used


(i) 1 2 3 .... n n(n 1) or 1 2 3 ... (n 1) n(n 1)
    

  
2 2




... (n 1) 

(ii) 1  2  3  .... n  n(n 1)(2n 1) or 1  2  3    2 n(n 1)(2n 1)
2
2
2
2
2
2
2
6 6

2

2

3

3

3
(iii) 1  2  3  .... n  n (n 1) 2 or 1  2  3  ... (n 1)  n (n 1) 2
3
3
3
3
3
4 4