This document discusses antiderivatives and basic integration rules and formulas: - The antiderivative of a function is the function whose derivative is the original function. Antiderivatives differ by a constant term. - Basic formulas are provided for finding antiderivatives of common functions like polynomials, trigonometric functions, and their inverses. - Integration rules allow for evaluating integrals of sums and products of functions using properties of antiderivatives. - Worked examples demonstrate applying formulas and rules to find antiderivatives of given functions.

1. Antiderivatives
5.1

2. Discovery of Power Rule for
Antiderivatives
If f ‘ (x) =
Then f(x) =
If f ‘ (x) =
Then f(x) =
If f ‘ (x) =
Then f(x) =
If f ‘ (x) =
Then f(x) =
3294 23
++− xxx
xxxx 33 234
++−
23 +− xx
xxx 2
2
3
3
2 22
3
+−
54
3
2
3 2
−+− x
x
x
xxxx 52
2
3
5
3 223
5
−++ −
534 2
++− xx
xxx 5
2
3
3
4 23
++
−

3. 1
oftiveAntideriva
1
+
=
+
n
x
x
n
n

4. Differentiation
Integration
The process of finding
a derivative The process of finding
the antiderivative
Symbols:
Symbols:
)(',', xfy
dx
dy
∫ dxxf )(
Integral
Integrand
Tells us the
variable of
integration

5. ∫ dxxf )( is the indefinite integral of f(x) with respect to x.
Each function has more than one antiderivative (actually infinitely many)
23
23
23
23
358
34
36
3
xx
xx
xx
xx
=−
=−
=+
=Derivative of:
The
antiderivatives
vary by a
constant!

6. General Solution for an Indefinite Integral
CxFdxxf +=∫ )()(
Where c is a constantYou will lose points if
you forget dx or + C!!!

7. Basic Integration Formulas
C
n
x
dxx
n
n
+
+
=
+
∫ 1
1
Ckxdxk +=∫
C
a
ax
dxax +
−
=∫
cos
)sin(
C
a
ax
dxax +=∫
sin
)cos(
C
a
ax
dxax +=∫
tan
)(sec2
C
a
ax
dxax +
−
=∫
cot
)(csc2
C
a
ax
dxaxax +=∫
sec
)tan(sec
C
a
ax
dxaxax +
−
=∫
csc
)cot(csc

8. Find:
∫ dxx5
∫ dx
x
1
∫ xdx2sin
∫ dx
x
2
cos
C
x
+=
6
6
CxCxdxx +=+== ∫
−
22 2
1
2
1
C
x
+
−
=
2
2cos
C
x
C
x
+=+=
2
sin2
2
1
2
sin
You can always check
your answer by
differentiating!

9. Basic Integration Rules
∫∫ = dxxfkdxxkf )()(
( ) ∫∫∫ ±=± dxxgdxxfdxxgxf )()()()(

10. Evaluate:
( )∫ +−+ dxxxx 465 23
∫ ∫∫∫ +−+= dxdxxdxxdxx 465 23
∫ ∫∫∫ +−+= dxdxxdxxdxx 465 23
CxC
x
C
x
C
x
++





+−





++





+= 4
2
6
34
5
234
Cxx
xx
++−+= 43
34
5 2
34
C represents
any constant

11. Evaluate:
∫ dxx5 3
∫= 5
3
x C
x
+=
5
8
5
8
Cx += 5
8
8
5

12. Evaluate:
( )∫ − dxxx cos3sin4
∫ ∫−= xdxxdx cos3sin4
CxCx +−+−= sin3cos4
Cxx +−−= sin3cos4

13. Evaluate:
( )∫ − dxx
22
3
( )∫ +−= dxxx 42
69
C
x
xx ++−=
5
29
5
3
Cxx
x
++−= 92
5
3
5

14. Evaluate:
∫ dx
x
x
2
cos
sin
∫ ⋅= dx
x
x
x cos
sin
cos
1
Cx += sec
∫ ⋅= xdxx tansec

15. Particular Solutions
2
1
)('
x
xf = and F(1) = 0
dx
x
xF ∫= 2
1
)(
dxxxF ∫
−
= 2
)(
C
x
xF +
−
=
−
1
)(
1
C
x
xF +−=
1
)(
0
1
1
)1( =+−= CF
1=C
1
1
)( +−=
x
xF