3.78 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS
Calculus
log (x^2 + 2x + 1) [Ans. x log (x^2 + 2x + 1) – 2x + xlog (x + 1+c)]
log x x 1 .( −^2 − )^ Ans.xlog x x 1 x 1 c( −^2 − +)^2 − + (^)
log x x a .( −^2 +^2 ) Ans.xlog x x a x a c( −^2 +^2 )−^2 +^2 + (^)
x 2
e 1 1
x x
(^) −
(^)
ex
(^) Ans. cx+ (^)
- x 2
e^11
x 1 (x 1)
(^) −
(^) + + (^)
ex
(^) Ans.x 1+ +c (^)
16. 2
1 1
logx−(logx)
Ans. x c
logx
(^) +
(^)
- x 9^2 + ( )
x x 9 9^22
Ans. 2 2 log x x 9 c
(^) +
- (^)
- 5 2x x− +^2 { }
(x 1) 5 2x x^22
Ans. 2 2log (x 1) 5 2x x c
(^) − − +
- − + − + +
(^)
- x 4^2 − Ans. x 4 2log(x x 4) c 2 x^2 − − +^2 − +
- 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