Calculus CC2053: 7
Introduction to Calculus and Linear Algebra 2011/2012 Semester 1
INTEGRATION: Anti-Derivative and Indefinite Integrals
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CC2053: Calculus 7
Antiderivative
A function F(x) for which
for every x in the domain f is said to be an antiderivative of f (x)
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CC2053: Calculus 7
Antiderivatives and Indefinite Integrals
is called the indefinite integral, represents the family of all antiderivatives of
if
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CC2053: Calculus 7
Rules for Integrating Common Functions
For k and C constants 1. The constant rule 2. The power rule 3. The logarithmic rule 4. The exponential rule for for all x 0 for
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CC2053: Calculus 7
Rules for Integrating Common Functions
For C constant 5. Sine function 6. Cosine function
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CC2053: Calculus 7
Example 1
Find the following integrals:
a.
b.
c.
d.
e.
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CC2053: Calculus 7
Example 1:- solution
a. b. c. d. e.
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CC2053: Calculus 7
Algebraic Rules for Indefinite Integration
6. The constant multiple rule
7. The sum and difference rule
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CC2053: Calculus 7
Example 2
Find the following integrals: a. c. e. b. d. f.
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CC2053: Calculus 7
Example 2:- solution
a. b.
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CC2053: Calculus 7
Example 2:- solution
c. d.
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CC2053: Calculus 7
Example 2:- solution
e. f.
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CC2053: Calculus 7
Example 3
Find the equation of the curve that passes through (0, 3) if its slope is given by for each x.
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CC2053: Calculus 7
Example 3:- solution
The curve passes through the point (0, 3). Substitute (0, 3) into
.
Thus, the equation of the curve is
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CC2053: Calculus 7
Example 4
If the marginal cost of producing x units is given by and the fixed cost is $2,000, find the cost function C(x) and the cost of producing 20 units.
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CC2053: Calculus 7
Example 4:- solution
Thus, the cost function is dollars and it costs $4200 to produce 20 units.
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CC2053: Calculus 7
Example 5
It is estimated that x months from now the population of a certain town will be changing at the rate of people per month. The current population is 5,000. What will be the population 9 months from now?
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CC2053: Calculus 7
Example solution
Current 5:- population is 5000, i.e. P(0) = 5000
At x = 9, Thus, 9 months from now, the population is 5,126 people.
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CC2053: Calculus 7
Integration by Substitution
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CC2053: Calculus 7
Reversing the Chain Rule
Recall:
Thus,
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CC2053: Calculus 7
Integration by Substitution
1. Introduce the letter u to stand for some expression in x that is chosen with the simplification of the integral as the goal. 2. Rewrite the integral in terms of u and du. Note: 3. Evaluate the resulting integral and then replace u by its expression in terms of x in the answer.
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CC2053: Calculus 7
Example 6
Find
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CC2053: Calculus 7
Example 6:- solution
Let Then (Step 1) (Step 2) (Step 3)
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CC2053: Calculus 7
Example 7
a. b.
c.
d.
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CC2053: Calculus 7
Example 7a:- solution
Let Then
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CC2053: Calculus 7
Example 7b:- solution
Let Then
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CC2053: Calculus 7
Example 7c:- solution
Let Then
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CC2053: Calculus 7
Example 7d:- solution
Let Then
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CC2053: Calculus 7
Example 8
The rate of change of the unit price p (in dollars) of shampoo is given by
where x is the number of bottles that supplier will make available in the market each week. a. Find the price-supply equation if the distributor of the shampoo is willing to supply 75 bottles a week at a price of $1.60 per bottle. b. How many bottles would the supplier be willing to supply at a price of $1.75 per bottle?
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CC2053: Calculus 7
Example 8:- solution
At p = 1.6, x = 75 i.e. Let Then
a. Thus, the price-supply equation is
b. At p = 1.75
The supplier will supply 125 bottles at price of $1.75.
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CC2053: Calculus 7
Integration by Parts
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CC2053: Calculus 7
Integration by Parts
If u and v are differentiable functions of x, then
Which can be written, in simpler form, as
Integration by parts may be useful when the integrand is a product of two functions u(x) v(x). This method works if the new product on the RHS is more easily integrable than the original product on the LHS.
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CC2053: Calculus 7
Example 9
Find
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CC2053: Calculus 7
Example 9:- solution
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