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MHT-CET : Mathematics Entrance Exam

MHT - CET : Mathematics - Definite Integrals Formulae Page 1

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 1. To express definite integral as the limit of a sum

 

Consider the sum Sn given by,

Sn = f(t1). d1 + f(t2). d2 + + f(tr). d6 + ..f(tn). dn =

f(tr). dr

This sum for the function 'f ' depends on 'n' and the choice of the points t1, t2, ..tn.

If for any choice of these points
t1, t2, ..tn and for any 'n' lim Sn exists, then this limit is called the definite integral of 'f ' on the interval [a, b] and is denoted by f(x)dx.

Thus, f(x)dx =

 

f(tr). dr

 

lim

n

d  0

 

 2. Formulae for definite integrals

1)

If f and g are integrable functions, then
[
f(x) + g(x)]dx = f(x)dx + g(x)dx

 

2)

If f and g are integrable functions, then
[
f(x) - g(x)]dx = f(x)dx - g(x)dx

 

3)

If f is an integrable function and 'k' is a constant, then
k f(x)dx = kf(x)dx

 

 3. Integration by parts

 

f(x) g(x)dx = [f(x). g(x)dx]- [f1(x) g(x)dx]dx

 

 4. Properties of Definite Integrals

 

  • f(x)dx = f(t)dt

  • f(x)dx = - f(x)dx

  • f(x)dx = 0

  • If a < c < b, then
    f(x)dx = f(x)dx + f(x)dx

  • f(x)dx = f(a - x)dx

  • f(x)dx = 2f(x)dx, if 'f ' is an even function

  • f(x)dx = 0, if 'f ' is an odd function.

  • f(x)dx = [f(x) + f(2a - x)]dx

  • f(x)dx = f(a + b - x) dx

 

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