FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.59(ii)
100 x100 1 x^101
x dx 100 1 101 c
+
= + = + (iii)x x1 1^2
xdx 1 1 2 c+
= + = +(iv) 1.dx = x + c
(v)
7 7 1^6
6
x dx x x^1 c
7 1 6 6x
− − + −
=− + =− =− + (see formula 3)(vi)
4/5 x^4 /5 1 x1/5 1/5
x dx 4 1 5x c
5 1 5− − +
= = = +
− +(vii) 3 x dx x dx x c^4 = 4 /3 = 73 7/3+
Note. (i) The arbitrary constant of integration (c) may be given in the final step.
(ii) Check:^5544
d 1x c 1 dx d 1c .5x 0 x
dx 5 5 dx dx 5
(^) + (^) = + = + =
(See Ex. 1)
(iii) So we find^4554
x dx x c,and^1 d 1x c x
5 dx 5
= +^ +^ =
(^)
Example 118 :(Integrate)
(i) 2x^4 dx (ii) (2x–3 + x^2 )dx
(iii) 3
3x x dx^1
x
(^) + −
(iv) (a + x)^2 dx
Solution:
(i)
4 4 x 24 1 5
2x dx 2 x dx (by B) 2.4 1 5x c
- = = + = +
(ii) (2x x )dx 2x dx x dx(byA)−^3 +^2 = −^3 + 2
3 2 x^31 x2 1 2 1 3
2 x dx x dx (by B) 2. 3 1 2 1 x 3 x c
− − + + −
= + = − + + + = − + +
(iii) 3x x dx^3 + −^1 x =3 x dx x dx 3 + 1 /2 − dxx (by A and B)
3x x^4 1/2 1
4 1 log x
2 1
= + −
3 2x x logx c. 4 3/2
=4 3+ − +