Paper 4: Fundamentals of Business Mathematics & Statistic

3.72 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

[Hint. I =

x/2

x/2

e dx

e 1

−

(^) − + , put e–x/2 + 1 = u & etc.]
(iii)
x 1 e 1
x e
e x dx
e x

−+ −

(^) +
Ans. log(e x )^1 x e
e
(^) +
(^)
(iv)
2 x
x
e dx.
(^) 2 e+ [Ans. x
x – 2 log (2 + ex)]
exdx udu x
(^) Hint.I= 2 x 2 u+ = + ,e u&etc.=

24. (i)
    xdx

(^) 2x 3+ ( ) ( )
Ans. 2x 3^1 3/2^3 2x 3
6 2
(^) − + − +
(ii)
3 x 2dx
x 1

+

(^) + ( )
(^) Ans.2 x 1 2 x 1+ 3/2− +

25. (i)

x (^1) dx
x 3

+

(^) + ( ) ( )
Ans. x 3 4 x 3^33 /2
2
(^) + − +
(^)
(ii)
x dx^2
(^) x 2+
Ans. x 2 52 ( + )^5 /2−^83 (x 2) 8(x 2)+ 3/2+ + 1/2

26. (i)
    4x 3dx
    2x 1

+

(^) −
(^) Ans. 2x 1 5 2x 1 32 ( − ) 3 /2+ − (^)
(ii)
1 xdx
1 x

−

(^) +
(^) Ans.4 x 1 x 1− + (^23) ( − )3/2 (^)
(iii)
1 xdx
1 x

−

(^) +
(^) Ans.4 x 1 1 x− − 32 ( + )3/2

27.

dx

(^) x x+ Ans.2log 1 x( )
(^) +
28.
dx dx
(^) x 1 x+ + { }
Ans. (1 x) x^2 3/2 3/2
3
(^) + −
(^)

29.

dx

(^) x 1 x+ − { }
Ans. (x 1) x^2 3/2 3/2
3
(^) + +
(^)

30.

dx

(^) x 2 x 3+ − + { }
Ans. (x 2) (x 3)^2 3/2 3/2
3

− + + +

(^)

31.

dx

(^) x 1 x 2+ + +
Ans. (x 1) (x 2)^2 3/2 3/2
3
(^) − + − +
(^)

3.5.3 STANDARD INTEGRALS

(A) 2 2

dx 1 x alog (x a)

x a 2a x a

= − >

(^) − +