Page 91 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes of continuity but it must also be true at most points when an integrable function is poorly
discontinuous.

Examples of Definite Integral Properties

1. The integral of y = 4x + 3 from 2 to 2 is 0 since the integral of any function in which both
bounds are the exact similar number is zero.
2. If the integral from 1 to 4 of the function y = x² is 21, then the integral from 4 to 1 of x²
is -21 since when the bounds of an integral are toggled, the integral is similar, but has the
opposite sign (positive transforms to negative, or negative transforms to positive).

Self Assessment

Fill in the blanks:
1. ............................... integral is where the integration is performed in a specific interval.

2. If the integration is performed between a to b, then a is known as lower limit and b is
known as ............................... limit.

3. For integration of definite integral we integrate the specified ............................... first then
apply the upper and lower limit.

4. The fundamental properties of integrals are simply attained for us since the integral is
defined straightforwardly by ............................... .
5. When a function involves a calculus integral on an interval it must also contain a calculus
integral on all ............................... .
6. When a function involves a calculus integral on two ............................... intervals it must
also contain a calculus integral on the amalgamation of the two intervals.
7. The integral on the large interval is the ............................... of the other two integrals.

8. Suppose f and g be integrable functions on the interval [a; b]. Subsequently f +g is also
b b
integrable on [a; b] and we have ............................... = a  f   x dx  a  g   x dx .

b b
9. The formula for ............................... is a  f   x dx  a  g   x dx if f(x)  g(x) for all but finitely
several points x in (a,b).
10. If both functions f (x) and g(x) contain a calculus integral on the interval [a,b] then any
............................... r f (x)+s g(x) (r, s  R) also encloses a calculus integral on the interval [a,b].
11. The integration by ............................... formula is defined by

b b
    x dx .
    x 
a  F   ´( )x G x dx  F x G a  ´ F x G
b g   b
    t dt 
12. The ............................... formula is defined by   f g t g ´   a  f   x dx .
a g
x d x
13. a  f   t dt is an ............................... integral of f and so, by definition, dx a  f   t dt  f   x .
c b
14. Complete the following: a  f   x dx  a  f   x dx  ............................... .

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