Math Grade XI

1. Learning Objectives
A student will be able to:
● Find antiderivatives of functions.
● Represent antiderivatives.
● Use basic antidifferentiation techniques.
● Use basic integration rules.

2. Integration / Antiderivatives

3. Integration / Antiderivatives

5. Look at the table:
Differentiation
F(x) F’(x)
Integration
3x2 + 3
3x2
3x2 - 5
3x2 + 5
6x
6x
6x
6x

6. If the constants 3,-5 and 5 is C ,then function F(x) = 3 x2 + C , with
then
1.2. Integral of
=
b. =
c.
=
by observing the order or pattern of the function above, if the coefficient of x is a
and the power of x is n, then in general it can be concluded
with n rational numbers and
a.
integral notation can be written

7. a.
d.
b.
c.
=
Determine the result of:
Answers:
a.
=
b.
=
=
e.
=
=
=
= =

8. =
=
=
=
=
d.
c.
e.

9. a.
Find indefinite integrals of:
f.
b. g.
c. h.
d. i.
e. j.

10. Remember Exponential Numbers:
1. 2.
4.
=
=
=
=
3.
3.a
3.b
4.a
4.b

11. =
b. =
c. =
=
d. =
=
=
e. =
=
=
f. =
=
=
a.
Answers :

12. g. h.
i.
j.
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=

13. = 2x + C
If 2 = a then = 2x + C can be written
1.a
2.a
2.b
If a = 1 then
If a = 1 then
Case.1
Case.2
Case.3
1.b
=
=
1.3. Determining the Basic Integral Formula:

14. Case conclusion 3
=
If 4 = k and it can be concluded
=
3.a
Ex :
20
=
20
20
=
=
=

15. =
=
=
=
+
3.b
Ex.1 :
+
+
+
+
+
=
=
=
=
C = C1+C2+…+Cn

16. Ex.2 :
= -
- +
=
=
=
Ex.3 :
Ex4:
=
=

17. a.
d.
b.
e.
c.
Determine the following indefinite integral
result!

18. a.
=
=
b. =
c.
=
=
=
=

19. =
e.
d.
=
=
=
=
=

20. 1.4. Substitution integral
If u = g (x) where g is a function that has a derivative
du
Derivative of u = Derivative of g(x)= g’(x)
Then f(u) = f(g(x))
=
=
=
= = =

21. Ex:
Find the integral of
Answer
: =
IF
Then derivative of
=
=
=
=

22. Find the integral of
Example :
If , then =
So,
=
=
=
.
= =
Example :

23. Another example :
Find the integral of
Answer:
If,