Paper 4: Fundamentals of Business Mathematics & Statistic

3.58 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

Example 114: ∫(x^4 ± x) dx = x^4 dx ± xdx

(B) A constant factor may be taken out from under the sign of integration and written before it. In

symbols, Audx A udx=

Example 115: ∫ (x^4 ± x) dx = ∫ x^4 dx ± (^) ∫xdx
(C) (A 1 u 1 ± A 2 u 2 - ±...........± An un) dx = A 1 u 1 dx ± A- 2 u 2 dx ± ..... ± An len dx.
Where A 1 A 2 , An are constants and u1, u 2 ..un are all functions of x.
Example 116: (x ± 3x^2 ) dx = xdx ± 3 x^2. dx.
Table of some fundamental integrals. The knowledge of differentiation will be employed now to find the
indefinite integral of number of function. The constant integration will be understood in all cases.
Farmulae :

1. n xn^1 d xn 1 n
   x dx (n 1) asn 1 dx n 1 x

+ +

= + ≠ − (^) + =

2. dx = x + c;


3. n n 1

dx (^1) ,(x 1)
(^) x (n 1)x= − − − ≠ + c (corr. of formula 1)

4. dxx=logx+ c

5.

mx emx x x

e dx= m (m 0)   corr, e dx e≠ =

6.

x x

e

a dx a

=log a^ + c (a > 0), a ±1)1)1)1)1)

7.

mx mx

e

a dx a

=m log a+ c, (a > 0), a ±1)1)1)1)1)

Standard Methods of Integration. The different methods of integration aim to reduce the given integral to

one of the above fundamental of known integral, Mainly there are two principle processes: (i) The method

of substitution, i.e , change of independent variable.

(ii) Integration by parts

If the integram is a rational fraction, it may be broken into partial fractions by algebra and then to

apply the previous method for integration.

Example 117:Integrate the following w.r.t.x.

(i) x^4 (ii) x^100 (iii) x (iv) 1 (v) –7 (vi) x–4/5 (vii)^3 x^4

Solution:

(i)

4 x^41

x dx 4 1 c

+

= + + (by Formulae 1) = 51 x c^5 +