The basics
For n
x n dx
1:
xn 1
C
n 1
Example:
x 2 dx
x3
3
C
Tips
. For
example:
3x 4 dx 3
x5
5
C
3x5
5
C
You can also break integrals up over addition and subtraction. In other
words, you can apply the rule to each piece of a function as long as the
pieces are added or subtracted.
For example:
5x
4
x5
2 x dx 5
5
x3
2
3
2
x
5
2 x3
3
C
If there is no exponent, it is assumed to be 1. For example:
x dx
x2
2
C since x
x1
The integral of a constant is that constant times the variable. For example:
3x C and
3 dx
5 dt
5t C
Finally, this rule applies no matter what the exponent is (except 1). It could
be a fraction, a decimal or a negative number! For example:
x
0.1
1.1
x
C and
1.1
1
4
x dx
1
1
4
x
C
1
1
4
3
1 4
4 4
x
1 4
4 4
C
5
4
x
5
4
C
4 54
x
5
C
Practice 1
Find the indefinite integral for each of the following functions.
1. 10x 4
4.
7.
1
x 2
2
3x
2.
5x 2
5.
x4
3x
10
3. 17x
8. 4 x0.2 3x0.5
1
2
2
10. x5
7
4 17
x
7
13. 2 x 2
x4
9
3 3
x
5
11. 12 x3 14 x5 1
14. 2 x1.5 1.8x2.4
4
6.
2
x 3x 2
5
9.
x2
1
3
15 13
12.
x 1
4
1
15.
x4
x4
Dealing with fractions and negative exponents
and subtraction.
The general rule is:
Therefore:
3 x
2
3
2
A B
C
A
C
B
C
x
4x 1
and
2
x2
4x
x2
1
x2
4
x
1
x2
When a variable in the denominator has an exponent, that can be rewritten using
a negative exponent.
The rule for this is: x
Therefore:
4
x
1
x2
1
4x
1
xn
n
x
2
and
5
x2
1
x3
5x
2
x
3
Once you have a function written this way, you can find the integral using the
power rule.
Example
3x 1
dx
x4
3x 1
3 1
dx
dx
4
4
x
x
x3 x 4
x31
x41
3
C
3 1
4 1
3
2
x
x
2
3
x
2
3x
3
x
4
dx
3
3
2
3x
3
C
C
The answer can also be written using fractions:
5
3
x
2
2
3x
3
C
3
2 x2
3
x3
C
Tips
Only the part with a negative exponent is written in the denominator when
rewriting your answer as a fraction.
For example:
2
x
5
2
2
5x2
The following are equivalent. You should check with your teacher to see
which version he or she prefers.
1 5
x
2
x5
2
Remember that a negative sign in the top or bottom of a fraction can be
brought out front.
2
3
2
3
2
3
Finally, and this is very important, while you can break a fraction up over
a single term in the denominator, you cannot break up over multiple terms!
2
x 5
2
x
2
3 x
and 2
5
x 2
3
x2
x
2
6
Practice 2
Find the indefinite integral for each function. Write your final answer as a fraction
or sum of fractions.
1.
x 5
x4
2.
4x x3
x3
3.
x2 x
x4
9 x 2 x5
2 x7
4.
x2 1
3x 2
5.
10 x 2 4
2
6.
7.
16 9x
x4
8.
2x 6
4
9.
10.
5 2x4
5x4
11.
13.
2 x2 x 1
4x4
14.
2x 1
x3
8 x6
x2
7
5x2
x 2
3x 4
12.
1 17x 2
x2
15.
12x5 x9
x
Fractional exponents and roots
Square roots, cube roots, and any other roots can be written using exponents.
This means that the power rule can be applied to these as well.
m
The general rule is:
n
xm
xn
1
1
Therefore:
x
x2 ,
3
3
x 3 , and
x
4
x3
x4 .
Using this, you can rewrite exponents before applying the power rule.
Example
3 x dx
1
2
1
3
1
x2
3x dx 3
1
1
2
x2
3
3
2
C
C
3
2 32
x
3
3
C
2x 2 C
Notice that most of the work is in the simplifying at the end!
Tips
If a root is in the denominator of a fraction, then the exponent will be
negative.
For example:
1
x
1
1
x
1
2
x2
Answers can be written with roots or with fractional exponents. This is
another thing you should check with your teacher, to see if one way of
writing the answer is preferred.
8
Practice 3
Find the indefinite integral of each of the following functions.
1.
2. 5 3 x
x 4
4. 2 x 4 3 x
5.
14
x
2
7 3 x2
8.
4
x
5
7.
10.
x2
1
x
13. 2 x5
3
x
14.
6.
13
x
5
1
x
11. 2
3
3. 2 3 x 2
4
5 x
9. 10 5 x 2 18 6 x
12.
4
2
4
3 x
9
2
x
3
15.
3
2
x
1
8 x
5
x
1
3
7 x2
Mixed practice
Use the following questions to test yourself with using the power rule for
integrals. All of the skills you practiced in this problem pack are covered.
For each of the following, find the indefinite integral.
3x 4 x
1.
10
4.
3
x
4
7.
1
x
8
4
x 5
3
10. x
13.
3
1
4
3
2 x2
1
x10
2.
5. 3x9
8.
x
1
3
4x 7
5 4
x
6
1
4
x2 6x
3.
x2 9 x 1
6.
13x 2 1
2 x2
9.
2x
11.
16 x3 4 x
5x
12.
14.
15 2
x
3
15.
23
x
3
10
2
3
x
5
x
2
7 x
5x 1
x12
Solutions
Practice 1
1. Answer: 2x5 C
x4 1
4 1
10 x 4 dx 10
10
5 3
x C or
3
2. Answer:
5 x 2 dx
x2 1
2 1
5
x5
5
2 x5 C
5 x3
3
C
5
x3
3
C
5 3
x C
3
C
(Note: answers like this can be written either with the fraction out front
or like
5 x3
3
3. Answer:
C . These are equivalent.)
17 2
x
2
17 x dx 17
x1 1
1 1
x2
4
4. Answer:
1 x1 1
2 1 1
x5 3x 2
50 2
x4
3x dx
10
x5 3x 2
50 2
6. Answer:
17 2
x
2
2x C
1 x2
2 2
2x C
x2
4
2x C
C
1 4
1 x4 1
x 3x dx
10
10 4 1
3
x1 1
1 1
C
1 x5
10 5
3
x2
2
C
x2
5
x3 C
C
x2
5
x3 C
2
x 3x 2 dx
5
7. Answer:
x2
2
2x C
1
x 2 dx
2
5. Answer:
17
2x
3
2
2 x1 1
5 1 1
x2 1
3
2 1
C
C
11
2 x2
5 2
x3
3
3
C
1
1
2
3
1
x2
3
1
1
2
3x dx
8. Answer: 4
x1.2
1.2
4 x0.2 3x0.5 dx
C
x1.5
1.5
C
x0.2 1
0.2 1
3
3
4
x2
3
3
2
C
x0.5 1
0.5 1
3
C
2 23
x
3
4
3
C
x1.2
1.2
2x 2
3
x1.5
1.5
C
C
(Note: although this could probably be simplified with a calculator or by
using fractions, it is OK to leave it in this form if it was given with
decimal exponents. If you are taking a class though, check to see what
2 32
x
3
9. Answer:
x
1
2
3x C
1
3 dx
x 6 1 87
10.Answer:
x
21 2
2 5
x
7
3
1
x2
1
1
2
3x C
9 dx
2 x5 1
7 5 1
9x C
7 x6
3
11.Answer: 3x 4
12.Answer:
x 6 1 87
x
21 2
8
1
4 x7
7 1 1
7
4 x7
7 8
7
9x C
9x C
x3 1
3 1
14
x5 1
5 1
x C
45 43
x
16
9x C
2 x6
7 6
3x C
x C
12 x3 14 x5 1 dx 12
7 x6
3
3x C
2 32
x
3
9x C
1
4 17
x
7
x 6 4 7 87
x
21 7 8
3x 4
x2
3
2
x C
12
x C 12
x4
4
14
x6
6
x C
1
4
1
15 13
15 x 3
x 1 dx
4
4 1 1
3
13.Answer:
2x
2
2 3
x
3
x
2 3
x
3
1 5
x
5
1 5
x
5
x
x C
45 34
x
16
x C
x5
5
3 x4
5 4
C
3 4
x C
20
x 2.5
2.5
1.8
x4 1
4 1
x5
5
1
4
4 54
x
5
3 x3 1
5 3 1
x3
2
3
C
C
1.8
x 2.4 1
2.4 1
C
2
x 2.5
2.5
1.8
x3.4
3.4
C
C
4 1
x
4 1
x dx
x3.4
3.4
x1.5 1
1.5 1
2 x1.5 1.8 x 2.4 dx 2
4
15 3 34
x
4 4
x C
3 4
x C
20
14.Answer: 2
15.Answer:
x C
x2 1
2
2 1
3 3
x dx
5
4
15 x 3
4 4
3
1
1
4
5
x
C
1
1
4
x
5
5
4
x
5
4
x5
5
C
4 54
x C
5
Practice 2
1. Answer:
1
2 x2
5
3x3
x 5
dx
x4
x
x4
5
dx
x4
C
x
3
5x 4 dx
x
2
2
5
x
3
3
C
1
2x2
5
3x 3
C
5 x3
5
(Note: the following are NOT the same
and
. Be sure that if there
3
3x 3
is a negative exponent, you move it to the denominator and make it
positive. It does not go into the numerator.)
2. Answer:
4
x
4 x x3
dx
x3
x C
4x
x3
x3
dx
x3
4x
2
1 dx
13
4x 1
1
x C
4
x C
x
3. Answer:
1
x
1
2 x2
x2 x
dx
x4
C
x2
x4
x2 1
dx
3x 2
x2
3x 2
5 x3
5. Answer:
3
5x
9
8x4
6. Answer:
x 3 dx
x1
1
x
2
2
1
2x2
C
1
dx
3x 2
1 1 2
x dx
3 3
1
1 x1
x
3
3 1
1
1
x
C
3
3x
C
x3
2 dx 5
3
2
2x C
5 x3
3
2x C
9x2
2 x7
x5
dx
2 x7
9
x
2
5
1 2
x dx
2
9 x4
2 4
1 x1
2 1
C
1
C
2x
16
3x3
7. Answer:
16 9 x
dx
x4
9
2 x2
16
x4
1 2
x
4
2x 6
dx
4
C
9x
dx
x4
16 x
4
3
9 x dx 16
x
3
3
9
x
2
2
C
3
x C
2
1
3
x
dx
2
2
5
3x
1
6 x2
2
9 x3
1 x2
2 2
3
x C
2
1 2
x
4
3
x C
2
C
5x2
x 2
5x2
x
2
5 2 1
dx
dx
x
x
4
4
4
4
3x
3x
3x 3x
3
3
5 x1
1 x2
2 x3
5
1
2
C
C
2
3 1
3 2
3 3
3x 6 x 9 x3
10.Answer:
1
x
C
1
C
2x
9 x 2 x5
dx
2 x7
9. Answer:
2
2x C
10 x 2 4
dx
2
8. Answer:
x
1
1
x
C
3
3x
4. Answer:
9
8x4
x
dx
x4
1
3x3
2
x C
5
14
3
2 4
x dx
3
16
3x3
9
2 x2
C
5 2 x4
dx
5x4
5
5x4
2 x4
dx
5x4
1
2 x2
C
2
x
11.Answer:
2x 1
dx
x3
2x
x3
1
dx
x3
2
2x
2
dx
5
4
x
3
x
2
x C
5
3
x 3 dx
2
x
1
x
1
1
3x3
2
x C
5
C
2
x
2
2
1
2x2
C
1
17x C
x
12.Answer:
1 17 x 2
dx
x2
1
2x
13.Answer:
17 x 2
dx
x2
1
x2
1
8x2
1
12 x3
x
x5
5
14.Answer:
8 x6
dx
x2
12 x5
15.Answer:
5
12 x5 x9
dx
x
x
17 dx
1
1
1
17 x C
x
17 x C
C
2 x2 x 1
2x2
x
dx
4
4
4x
4x 4x4
1 x1
1 x2
1 x3
2 1
4 2
4 3
8
x
2
1
dx
4x4
C
1 2 1 3
x
x
2
4
1
1
1
2
2 x 8 x 12 x3
1 4
x dx
4
C
C
8
x2
x6
dx
x2
x9
9
C
12 x5
x
8x
x9
dx
x
2
12 x
4
x dx 8
x
1
1
x5
x dx 12
5
4
8
x5
5
C
x9
9
x5
5
C
12 x5
5
x9
9
8
x
C
C
Practice 3
1. Answer:
2 3
x
3
x 4 dx
x
4x C
1
2
4 dx
1
1
2
x
4x C
1
1
2
15
3
2
x
3
2
4x C
2 23
x
3
4x C
2 3
x
3
4x C
15 43
2. Answer:
x
4
1
1
3
5 x dx
x3
5
4
3
C
15 43
x
4
C
C
6 53
x C
5
3. Answer:
2
2
3
2 3 x 2 dx
4
4 32
x 3x 3
3
4
4 32
x 3x 3
3
6. Answer:
2
x dx
3
x3
2
5
3
C
6 53
x C
5
C
C
1
3
1
2
4 x dx
1
1
1
x2
2
1
1
2
1
x3
4
1
1
3
C
3
4
x2
2
3
2
x3
4
4
3
C
2 54
5. Answer: x
5
14
x dx
2
2x
5
1
x3
2
2
1
3
2 x dx
2 x 4 3 x dx
7. Answer:
4
1
x3
5 x dx 5
1
1
3
3
4. Answer:
C
C
1
4
1
x dx
2
4 32
x
9
1
1
4
1 x
2 1 1
4
C
5
4
1 x
2 5
4
2 54
x C
5
C
C
1
2
2
x dx
3
21 53
x
5
1
1
2
2 x
3 1 1
2
C
1 3
x C
3
16
3
2
2 x
3 3
2
C
4 32
x C
9
C
7 3 x2
x 2 dx
21 53
x
5
7x
13
x dx
5
8 32
x
15
3 43
x
20
9. Answer:
10 x
x3
3
x3
3
x3
7
5
3
C
C
4
x
5
C
1
x dx
5
6
x
18
7
6
10.Answer: 2x
1
dx
x
x
1
dx
4
x
12.Answer: 3x
C
4 x
5 3
2
2
1
6
1
1
x5
18 x dx 10
2
1
5
50 75
x
7
108 76
x
7
1
x6
18
1
1
6
C
C
C
1
2
1
dx
10 x
1
1
x 2
C
1
1
2
x2
1
2
1
C
2x 2
C
C
1
4
2 x
2
3
2
5
C
4 34
x
3
11.Answer: 2 x
1 x
5 1 1
3
C
10 x
7
6
1
2
4 x
5 1 1
2
3
2
C
18 x dx
x
10
7
5
1
1
3
1
1
2
1
3
1
2
50 75 108 76
x
x
7
7
7
5
2
x 2 dx
3 43
x
20
4
x
5
2
5
1
x3
7
2
1
3
1 3
x C
3
8 32
8. Answer:
x
15
5
2
2
3
1
2
dx
2x
1
1
4
x
C
1
1
4
C
17
2x
3
4
x
3
4
C
4 43
2x
x C
3
4
3
1 x
5 4
3
C
2
3
x
5
dx
x
2
2x
1
3
1
2
5x
1
dx
2
1
1
x 3
1
1
3
5
1
x 2
1
1
2
C
2
1
x3
2
2
3
x2
5
1
2
C
1
3x 3 10 x 2
C
9 23
x
2
4 72
x
7
13.Answer:
2 x5
3
dx
3
x
4 72
x
7
9 23
x
2
2x
2
4
5 x
3 x
8 12
x
5
8 34
x
9
15.Answer:
5
2
3x
1
3
5
1
1
x2
2
5
1
2
dx
1
x 3
3
1
1
3
C
7
2
x2
2
7
2
x3
3
2
3
8 34
x
9
C
4
x
5
dx
1
2
2
x
3
1
4
dx
4
5
1
1
2
x
1
1
2
1
1
4
2
3
x
1
1
4
1
7
x 3
2
1
3
C
1
2
4 x
5 1
2
2 x
3 3
4
C
3 13
x C
7
1 12
x
4
1
8 x
7 3 x2
1 12
x
4
3 13
x C
7
1
x
8
dx
1
2
1
x
7
2
3
1
dx
1
8
2
1
x 2
1
1
2
1
C
1
1
1 x2
8 1
2
1 x3
7 1
3
Mixed Practice
3 5
x
50
3x 4 x
dx
10
3
4
C
1
1. Answer:
C
C
8 12
14.Answer: x
5
4
C
1 2
x C
20
3x 4
10
x
3 x5
dx
10
10 5
1 x2
10 2
18
C
3 5
x
50
1 2
x C
20
C
2. Answer:
1
x10
1
9 x9
1
x C
4
1
dx
4
1 3
x
3
3. Answer:
1
dx
4
10
x
9 2
x
2
x3
3
3
8x2
1
24 x3
3
x
4
3
5. Answer:
3x9
3 10
x
10
9
x2
2
9
1
9 x9
1
x C
4
x C
1 3
x
3
9 2
x
2
x C
C
3 x2
4 2
1 4
x dx
8
1
x C
4
x C
x 2 9 x 1 dx
4. Answer:
9
x
1 x3
8 3
C
3
8x2
x2
2
C
1
24 x3
C
1 3
x 3x 2 C
3
x 2 6 x dx 3
x10
10
x3
3
6
3 10
x
10
1 3
x 3x 2 C
3
13
1
x
C
2
2x
6. Answer:
13x 2 1
dx
2x2
13x 2
2x2
1
dx
2x2
13
1 x1
x
2
2 1
13 1 2
x dx
2 2
C
13
1
x
C
2
2x
7. Answer:
2 32
x
9
x 5
dx
3
8. Answer:
5
x C
3
1
x
3
1 5
x
6
5 4
x 2 x dx
6
2
1
2
5
dx
3
1
1
2
1 x
3 1 1
2
5
x C
3
1 x
3 3
2
x2 C
5 x5
6 5
2
x2
2
C
1 5
x
6
1
9. Answer: 3x 3 10 x 2
C
19
3
2
x2 C
5
x C
3
2 23
x
9
5
x C
3
2
3
x
3x
5
dx
x
2
3
10 x
x
x
11.Answer:
3 23
x
2
1
1
3
16 3
x
15
7 x
3
2 x2
4 12
x
7
2
x
7
C
1
x2
5
1
2
C
1
x 3
C
1
1
3
3
2
x4
3
4
x3
2
3
4 43
x
3
3 32
x C
2
16 x3
5 3
4
x C
5
C
4x
dx
5x
16 2
x
5
4
dx
5
16 3
x
15
4
x C
5
1
1
2
2
7
dx
1
1
x 2
1
1
2
2 x2
7 1
2
C
4 12
x
7
C
C
3
2 x2 7 x C
2x
5 75
x
21
23
x dx
3
5 75
x
21
1 43
x
2
1
2 x10
3
x
2
1 43
x
2
15 2
x
3
15.Answer:
5
2
x3
2
2
3
C
4 x 7 dx
14.Answer:
1
1
16 x3
5x
dx
13.Answer:
2
1
x 2
1
1
2
4
x C
5
16 x3 4 x
dx
5x
2
dx
1
1
x 3
1
1
3
C
x 4
1
1
4
dx
12.Answer:
5x
1
1
2
C
4 34
x
3
10.Answer:
1
4
1
2
2x
1
3
2
4 x 7 dx
3 x1
2 1
x2
2
7x C
3
2x2 7 x C
2x
C
1 52
x
3
2 13
x dx
3
2
1
1 x5
3 2 1
5
C
1
11x11
4
C
20
1
1
2 x3
3 1 1
3
C
7
4
1 x5
3 7
5
2 x3
3 4
3
C
5x 1
dx
x12
5x
x12
1
dx
x12
5x
11
x
12
dx 5
21
x
10
10
x
11
11
C
1
2 x10
1
C
11x11