Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.63

7. (i) x2xdx (ii) e-4xdx (iii) e–mxdx (iv) e2logx (v) emlog x dx

e^2 x e4x emx x^3 xm 1

Ans. (i) (ii) (iii) (iv) (v) 2 4 m 3 m 1

− +

(^) − + (^)

8. (i) 5 x dx (ii) 5 2x dx

x 2x

e e

Ans. (i)^5 (ii)^5

log 5 2log 5

(^)

9. (i)

4 x 2x

3x

e e dx

e

+

(ii)

2 x x

x

e e 1dx

e

+ +

(iii)

3 x x

x x

e e dx

e e−

+

(^) +
(iv)
2x 4x
x
3e 3e dx
e e

+

(^) + (v)
2x
x x
1 e dx
e e−

+

(^) +
x x x x e3x 3x x
(^) Ans. (i) e e (ii) e x e (iii) (iv) e (v) e− − + − − 2
Evaluate:

10. (i)
    3 x 5 dx^2
    x

(^) + +
Ans.2x 5x 2logx c3/2+ + +^
(ii) ( )
1 x^3
−x dx
3 2 x^3
Ans.logx 3x x− +2 3− +c^

11. If dydx=x^3 + x + x + 1 and if y = 2 when x = 1, find the value only.

x x x^4321

Ans.4 3 2+ + +x− 12

12. If f′(x)

3x 5x

x x

e e

e e−

+

+ and f(0) =

1

4 , find f(x).

e^4 x

(^) Ans. 4
4x x x 4x
x x
Hints.f (x) e (e e e ,f (0) 1 1 c,c 0&etc.
e e 4 4
−
−
(^) ′ = + = ′ = = + =
(^) +

13.

x 1 e 1

x e

e e dx

e x

−+ −

(^) +
Ans. log(e x^1 x e
e
(^) +
(^)
x 1 e 1 x e 1 x e
x x x e
Hint.I 1 e.e ex dx 1 e ex dx 1 du,u e x &etc.
e e e e e e e u

− + − + −

= + = + = = +

3.5.1 METHOD OF SUBSTITUTION

Let I (x)dx then f(x)= (^) dxdI=