Paper 4: Fundamentals of Business Mathematics & Statistic

3.78 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

11. 
    

    log (x^2 + 2x + 1) [Ans. x log (x^2 + 2x + 1) – 2x + xlog (x + 1+c)]

    
    


12. 
    

    log x x 1 .( −^2 − )^ Ans.xlog x x 1 x 1 c( −^2 − +)^2 − + (^)

    
    


13. 
    

    log x x a .( −^2 +^2 ) Ans.xlog x x a x a c( −^2 +^2 )−^2 +^2 + (^)

    
    


14. 
    

    x 2
    e 1 1
    x x

    
    

(^) −
(^)
ex
(^) Ans. cx+ (^)

15. x 2
    e^11
    x 1 (x 1)

(^) −
(^) + + (^)
ex
(^) Ans.x 1+ +c (^)

16. 2

1 1

logx−(logx)

Ans. x c

logx

(^) +
(^)

17. x 9^2 + ( )

x x 9 9^22

Ans. 2 2 log x x 9 c

(^) +

• 
  
  
  
  • 
    
    
    
    • 
      
      
      
      • (^)
      
      
      
      
    
    
    
    
  
  
  
  

18. 5 2x x− +^2 { }

(x 1) 5 2x x^22

Ans. 2 2log (x 1) 5 2x x c

(^) − − +

• − + − + +
  (^)

19. x 4^2 − Ans. x 4 2log(x x 4) c 2 x^2 − − +^2 − +


20. 4x 4x 10.^2 − + Ans.2x 1 4 − 4x 4x 10 log 2x 1 4x 4x 10 c^2 − + + 49 { − +^2 − + }+^

3.5.5 DEFINITE INTEGRALS

Definition:

Let a function f(x) has a fixed finite value in [a, b] for any fixed value of x in that interval i.e., for a # x # and

f(x) is continuous in [a, b], where a and b both are finite, (b>a).

Let the interval [a, b] be divided in equal parts having a length h. Now the points of division (on x axis) will

be.

x = a +h, a +2h ..., a + (n – 1) h, b – a = nh.

Now limh f(a h) f(a 2h) ...... f(a nh)]h 0→ + + + + + +^

i.e.,

n

h 0limh f(a rh)→^ r 1= + , (if it exists) is called definite integral of the function f(x) between the limits a and b and