The document discusses various techniques for evaluating indefinite integrals (antiderivatives), including: 1) Using power rules to evaluate basic integrals like ∫ 4x3 dx = x4 + C 2) Expanding rational or negative exponents before integrating 3) Expanding expressions before integrating term by term 4) Simplifying rational expressions by factoring and canceling before integrating 5) Setting up word problems involving integrals to find related functions like total cost, revenue, distance over time.

1. 11.1 Antiderivatives and
Indefinite Integrals
A. Let’s figure out the formula.
B. Those with rational or negative
exponents
C. Those that need expanding
D. Those that need simplifying
E. Word Problems

2. A. Let’s figure out the formula.
Recall the power rule for derivatives went like this :
x n → nx n −1
Consider these basic ones :
x 4 + 13 → 4 x 3 x 4 + 7 → 4 x3 x 4 + 122 → 4 x 3
Since it doesn' t matter what the constant term is, I could say :
x 4 + C → 4 x 3 , where C is any constant.
Our next task will be trying to take the 4 x 3 and get the
" indefinite integral" of it x 4 + C.

3. How can we get from 4 x to x + C ?
3 4
 First add one to its exponent.
 Then, divide the term by its new exponent.
 Last, add the + C.
3+1
4x
4x →
3
+C
3 +1
4
4x
= +C
4
= x +C
4

4. This is sort of undoing differentiation!
• It’s like we are answering the riddle: “What is the
function whose derivative is this?”
∫ 4 x dx = " indefinite integral of 4 x = x +C
3 3 4
n +1
x
∫ x dx = n + 1 + C
n

5. ∫ 6 x dx You try : ∫ 12 x dx
2 3

6. Term by term :
∫(x )
+ 4 x 3 − 3 x 2 + 7 dx
5
is the same as ∫ x 5 dx + ∫ 4 x 3dx − ∫ 3 x 2 dx + ∫ 7 dx
i.e., take the integral of each term, like 4 smaller tasks.
Notice the last term is ∫ 7 dx. What function would have
a derivative of just plain 7? Answer 7x.
x 6 4 x 4 3x 3
+ − + 7x + C
6 4 3
x6
= + x 4 − x3 + 7 x + C
6

7.  8 5
2x 
You try : ∫  3x − 2 x + x −

7 6
+ 14 dx

 3 

8. B. Those with rational or negative
exponents
• As you have seen before, sometimes it is
necessary to REWRITE the problem in
exponential form without negative exponents
BEFORE using the rules.
• Examples of rewriting:

9.  3  4 3 6  dx
∫  4 x − x 2 dx
 
∫  3 ⋅ x + 4 x 3 dx




∫ x3

10. You try :
 7  3 2 2  du
∫

6 t − 2 dt
t  ∫
4⋅ x +

3
x2
dx


∫ u4

11. C. Those that need expanding
• You know that you can integrate term-by-
term. Sometimes you have to EXPAND things
(F.O.I.L., distribute, etc.) before you can do
the term-by-term thing.
• “Terms” are only connected to each other by
plus signs. Nothing else.

12. ∫ ( x + 2) 3 dx You try : ∫ ( x − 1) dx
3

13. ∫ ( x + 2) 2 x dx You try : ∫ ( x − 1)
23
x dx

14. 4 x 4 − x 3 + 6 x 2 + 14 x 5x5 + x 4 − 4 x 2 + 6 x
∫ x
dx You try : ∫
x
dx

15. D. Those that need simplifying
( x + 3)( x + 5) dx = ( x + 3) dx = x 2
∫ x+5 ∫ 2
+ 3x + C
•FACTOR TOP AND BOTTOM
•CANCEL VERTICALLY IF POSSIBLE
•INTEGRATE TERM BY TERM

16. 3x − x − 2
2
You try : ∫ dx
x −1

17. E. Word Problems
Differentiation does this :
profit/revenue/cost → marginal profit/revenue/cost
curve → slope of the tangent line
distance → velocity
velocity → acceleration
any quantity → rate of change
So Integration does the opposite :
marginals → profit/revenue/cost functions
slope of the tangent line → the equation of the curve
velocity → distance
acceleration → velocity
rate of change → the function for the quantity

18. A company's marginal cost function is MC ( x ) = 6 x and the
fixed cost is $1000. Find the cost function.

19. A company's marginal revenue function is MR = 12 x + 83 x ,
where x is the number of units. Find the revenue function.
[Hint for finding C : How much revenue is brought in for zero
production? zero.]

20. 3
A person can memorize words at the rate of words per minute.
t
Find a formula for the total number of words that can be memorized
in t minutes. [Hint for finding C : How many words can be memorized
in zero minutes? zero.]

21. dC
The marginal cost for producing x units of a product is modeled by = 32 − 0.04 x.
dx
It costs $50 to produce one unit. Find the total cost of producing 200 units.