Try

1. Integration
• Most operations in mathematics have an
inverse operation.
• Addition – Subtraction
• Multiplication – Division
• Is there an inverse operation for
differentiation?
• INTEGRATION.

2. • f(x) = x3
• g(x) = 3x2
• Sometimes, the derivative of a function is
known, and we want to find the original
function.
• What is the function, whose derivative is 3x2
?
• Anti-derivative of a given function.
• Given a function g(x), its anti-derivative is f(x)
• f(x) =  g(x)dx
• The sign  is integral sign.
• g(x) is the integrand
• dx indicates that we are integrating w.r.t. x.

3. What does integration mean?
• The instantaneous speed of rocket is 0.2t
km/sec at t seconds after it is launched.
• How much distance has the rocket covered
after traveling for 5 seconds?
• Speed is not fixed, but is a function of time.
• Let t = time
• y = distance,
• then what is speed?
• Speed = change in distance/ change in time
• Speed = dy/dt.

4. • Given dy/dt, find y(t).
• Given f’(t) find f(t). INTEGRATION.
• If dy/dt = 0.2t, what is the original function,
whose derivative is 0.2t?
• We know that 2t = d(t2
)/ dt .
• So, 0.2t = 0.1*2t = d(0.1t2
)/ dt.
• y(t) = distance covered in t seconds
• = 0.1t2
Km.
• distance covered in 5 seconds = 2.5 Km.

5. Application: Advertising
A satellite radio station is launching an aggressive advertising
campaign in order to increase the number of daily listeners.
The station currently has 27,000 daily listeners, and
management expects the number of daily listeners, S(t), to
grow at the rate of
listeners per day, where t is the number of days since the
campaign began. How long should the campaign last if the
station wants the number of daily listeners to grow to 41,000?
Ans. 50 days (approx.)

6. Constant of Integration
• f(x) = 5x2
, what is f’(x)?
 f’(x) = 10x
• f(x) = 5x2
+ 10, what is f’(x)?
 f’(x) = 10x
• f(x) = 5x2
+ 50, what is f’(x)?
 10x Again.
• What is the anti-derivative of 10x?
 5x2
Or
 5x2
+ 10 Or
 5x2
+ 50?
• Add any arbitrary constant to 5x2
, the derivative is the
same.
• Anti-derivative of 10x is 5x2
+ C.
• C is called constant of integration.

7. Some Common Integration Rules
c
e
a
dx
e ax
ax



1

8. Cost
• If the total cost of producing and marketing
x units of a commodity is C(x)
• Then average cost per unit is C(x)/x
• And the marginal cost is C’(x)
• Thus the total cost = ∫ marginal cost dx
= ∫ C’(x) dx =C(x) + c
• Example:
• Marginal cost C’(x) = 1.064 – 0.005x,
• x is the number of units manufactured.
• Find the total cost for producing 4 units if
the fixed cost is 16.3.

9. Application: Cost Function
If the marginal cost of producing x units of a commodity is
given by
C’
and the fixed cost is $2,000, find the cost function C(x) and the
cost of producing 20 units.
Ans. $3200

10. Application: Cost Function
A manufacturer has found that marginal cost is
C’
dollars per unit when q units have been produced. The total cost
of producing the first 2 units is $900. What is the total cost of
producing first 5 units?
Ans. $1587

11. Application: Storage Cost
A retailer receives a shipment of 10,000 kilograms of rice that
will be used up over a 5-month period at the constant rate of
2,000 kilograms per month. If storage costs are 1 cent per
kilogram per month, how much will the retailer pay in storage
costs over the next 5 months?
Ans. $250

12. Revenue
• R(x) is the total revenue.
• x is the demand.
• Marginal Revenue:
• R’(x)=dR(x)’/dx.
• Total Revenue ∫R’(x) dx = R(x) + c.
• Example:
• If the marginal revenue function is
R’(x)=8 – 6x – 2x2
• Determine total revenue.

13. Application: Revenue Function
Find the revenue function R(x) when the marginal revenue is
R’
and no revenue results at a zero production level. What is the
revenue at a production level of 1,000 units?
Ans. 200,000

14. Application: Marginal Profit
The marginal profit of a certain commodity is
P’
When q units are produced. When 10 units are produced, the
profit is $700.
(A)Find the profit function P(q).
(B)What production level q results in maximum profit? What is
the maximum profit?
Ans. (A)
(B) Production results maximum at q=50 and
maximum profit is $2300.

15. Integration by Substitution
• Select a substitution that appears to simplify
the integrand.
• Select u so that u’ is a factor in the
integrand.
• Express integrand in terms of u and du,
completely eliminating x and dx
• Find the integration w.r.t. u
• Substitute u=u(x)
• Examples:

16. Generalized Power Rule
    c
x
u
b
dx
x
u
x
u
b
b



 

1
)
(
1
1
)
(
)
(

17. Integration by parts
  c
dx
dx
x
g
x
f
dx
x
g
x
f
dx
x
g
x
f 



    )
(
)
(
)
(
)
(
)
(
)
(
 dx
x
xsin

18. Integration by partial fractions
• Rational function is transformed into the
sum of simpler functions.
• Method is used for proper fractions
• Polynomial in numerator is of lower degree
than the polynomial in denominator.
• We get the functions in following forms:
• A/ (ax+b)
• (Ax+B)/ (ax2
+bx+c)

19. • Some forms of partial fractions:
• We have to obtain the values of A, B, C,
• Multiply both sides by LHS denominator
• Equate the multiples of the same powers of x.
3
2
3
2
2
)
(
)
(
)
(
)
(
)
(
)
)(
(
)
(
)
)(
(
)
(
a
x
C
a
x
B
a
x
A
a
x
x
f
h
gx
C
c
bx
ax
B
Ax
h
gx
c
bx
ax
x
f
b
x
B
a
x
A
b
x
a
x
x
f























20. • Example:
• We can write the integrand as
• Now, we obtain A, B and C
• Comparing the powers of x, we get
• A+B = 1, 2A+B = 3
• Solving these equations, we get A=2, B = -1
dx
x
x
x
 


2
3
3
2
2
1
)
2
)(
1
(
)
3
(
2
3
3
2











x
B
x
A
x
x
x
x
x
x
)
2
(
)
(
3
or
)
1
(
)
2
(
)
3
(
B
A
B
A
x
x
x
B
x
A
x











21. • Thus


 






dx
x
dx
x
dx
x
x
x
2
1
1
2
2
3
3
2
)
2
ln(
)
1
ln(
2 


 x
x
)
2
(
)
1
(
ln
2



x
x