An integral can be defined for all values (that is, for all X X ). An example of this type of function is the polynomial - which we will study in the coming years.

However, there are integrals that are not defined for all values (all X X ), since if we place certain X X or a certain range of values of X X we will receive an expression considered "invalid" in mathematics. The values of X X for which integration is undefined cause the discontinuity of a function.

integrals that are not defined for all values

Suggested Topics to Practice in Advance

  1. Ways to Represent a Function
  2. Representing a Function Verbally and with Tables
  3. Graphical Representation of a Function
  4. Algebraic Representation of a Function
  5. Notation of a Function
  6. Rate of Change of a Function
  7. Variation of a Function
  8. Rate of change represented with steps in the graph of the function
  9. Rate of change of a function represented graphically
  10. Constant Rate of Change
  11. Variable Rate of Change
  12. Rate of Change of a Function Represented by a Table of Values
  13. Functions for Seventh Grade
  14. Increasing and Decreasing Intervals (Functions)
  15. Increasing functions
  16. Decreasing function
  17. Constant Function
  18. Decreasing Interval of a function
  19. Increasing Intervals of a function

Practice Indefinite integral

Examples with solutions for Indefinite integral

Exercise #1

Does the given function have a domain? If so, what is it?

9x4 \frac{9x}{4}

Video Solution

Step-by-Step Solution

Since the function's denominator equals 4, the domain of the function is all real numbers. This means that any one of the x values could be compatible.

Answer

No, the entire domain

Exercise #2

Given the following function:

5x \frac{5}{x}

Does the function have a domain? If so, what is it?

Video Solution

Step-by-Step Solution

Since the unknown is in the denominator, we should remember that the denominator cannot be equal to 0.

In other words, x0 x\ne0

The domain of the function is all those values that, when substituted into the function, will make the function legal and defined.

The domain in this case will be all real numbers that are not equal to 0.

Answer

Yes, x0 x\ne0

Exercise #3

Given the following function:

5+4x2+x2 \frac{5+4x}{2+x^2}

Does the function have a domain? If so, what is it?

Video Solution

Step-by-Step Solution

Since the denominator is positive for all X, the domain of the function is the entire domain.

That is, all X, therefore there is no domain restriction.

Answer

No, the entire domain

Exercise #4

Given the following function:

65(2x2)2 \frac{65}{(2x-2)^2}

Does the function have a domain? If so, what is it?

Video Solution

Step-by-Step Solution

The denominator of the function cannot be equal to 0.

Therefore, we will set the denominator equal to 0 and solve for the domain:

(2x2)20 (2x-2)^2\ne0

2x2 2x\ne2

x1 x\ne1

In other words, the domain of the function is all numbers except 1.

Answer

Yes, x1 x\ne1

Exercise #5

Consider the following function:

3x+42x1 \frac{3x+4}{2x-1}

What is the domain of the function?

Video Solution

Answer

x12 x\ne\frac{1}{2}

Exercise #6

Given the following function:

128x4 \frac{12}{8x-4}

What is the domain of the function?

Video Solution

Answer

x12 x\ne\frac{1}{2}

Exercise #7

Given the following function:

2421x7 \frac{24}{21x-7}

What is the domain of the function?

Video Solution

Answer

x13 x\ne\frac{1}{3}

Exercise #8

Given the following function:

235x2 \frac{23}{5x-2}

Does the function have a domain? If so, what is it?

Video Solution

Answer

Yes, x25 x\ne\frac{2}{5}

Exercise #9

Given the following function:

49+2xx+4 \frac{49+2x}{x+4}

Does the function have a domain? If so, what is it?

Video Solution

Answer

Yes, x4 x\ne-4

Exercise #10

Given the following function:

5x2x \frac{5-x}{2-x}

Does the function have a domain? If so, what is it?

Video Solution

Answer

Yes, x2 x\ne2

Exercise #11

Look at the following function:

10x35x3 \frac{10x-3}{5x-3}

What is the domain of the function?

Video Solution

Answer

x35 x\ne\frac{3}{5}

Exercise #12

Look at the following function:

2010x5 \frac{20}{10x-5}

What is the domain of the function?

Video Solution

Answer

x12 x\ne\frac{1}{2}

Exercise #13

Look at the following function:

2x+202x10 \frac{2x+20}{\sqrt{2x-10}}

What is the domain of the function?

Video Solution

Answer

x > 5

Exercise #14

Look at the following function:

2x+23x1 \frac{2x+2}{3x-1}

What is the domain of the function?

Video Solution

Answer

x13 x\ne\frac{1}{3}

Exercise #15

Look at the following function:

2x+29x+6 \frac{2x+2}{9x+6}

What is the domain of the function?

Video Solution

Answer

x23 x\ne-\frac{2}{3}

Topics learned in later sections

  1. Domain of a Function
  2. Inputing Values into a Function