Indefinite integral

🏆Practice domain of a function

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

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Test yourself on domain of a function!

\( \frac{6}{x+5}=1 \)

What is the field of application of the equation?

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  • An example of this is a function with a fraction with values X X in the denominator.
  • For example 1x1\over x
    According to mathematical rules, the denominator of a fraction cannot be zero since it is not possible to divide by zero. Therefore, when there is a possibility that the denominator equals zero, the integral cannot be defined for the values of X X that could cause the denominator to be zero.
Indefinite Integral
  • Another example is a square root function. For example
    According to the algebraic rules, the expression under the square root cannot be negative, that is, it must be positive or zero, but in no way negative. Therefore, The integral will be undefined for a range of values of X X that cause the expression under the square root to be negative.f(x)=x2x5f(x)=\sqrt{x^2-x-5}
Example of a negative square root function


Examples and exercises with solutions of 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

Look at 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 variable 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

Look the following function:

15x4 \frac{1}{5x-4}

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 15x4 \frac{1}{5x-4} , we need to find the values of x x for which the function is undefined. This occurs when the denominator equals zero:

First, set the denominator equal to zero:
5x4=0 5x - 4 = 0

Next, solve for x x :
5x=4 5x = 4
x=45 x = \frac{4}{5}

The function is undefined at x=45 x = \frac{4}{5} . Therefore, the domain of the function includes all real numbers except x=45 x = \frac{4}{5} .

In mathematical notation, the domain is:
x45 x \ne \frac{4}{5} .

This matches choice 3 among the given options.

Answer

x45 x\ne\frac{4}{5}

Exercise #4

Look at the following function:

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

What is the domain of the function?

Video Solution

Step-by-Step Solution

To determine the domain of the function 2010x5 \frac{20}{10x-5} , we need to ensure that the denominator is not zero.

  • Step 1: Identify the denominator, which is 10x5 10x - 5 .
  • Step 2: Set the denominator equal to zero and solve for x x . This gives us the equation:

10x5=0 10x - 5 = 0

  • Step 3: Add 5 to both sides of the equation:

10x=5 10x = 5

  • Step 4: Divide both sides by 10 to isolate x x :

x=510 x = \frac{5}{10}

  • Step 5: Simplify the fraction:

x=12 x = \frac{1}{2}

This means that the function is undefined at x=12 x = \frac{1}{2} . Therefore, the domain of the function is all real numbers except x=12 x = \frac{1}{2} .

Therefore, the domain of the function is x12 x \ne \frac{1}{2} .

Answer

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

Exercise #5

Given the following function:

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

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

Video Solution

Step-by-Step Solution

To determine the domain of the function 235x2 \frac{23}{5x-2} , follow these steps:

  • Step 1: Identify where the function is undefined by setting the denominator equal to zero.
    Equation: 5x2=0 5x - 2 = 0
  • Step 2: Solve the equation for x x .

Let's perform the calculation:
Step 1: Set 5x2=0 5x - 2 = 0 .

Step 2: Solve for x x by adding 2 to both sides:
5x=2 5x = 2

Next, divide both sides by 5:
x=25 x = \frac{2}{5}

This shows that the function is undefined at x=25 x = \frac{2}{5} , thus excluding this point from the domain of the function.

The domain of 235x2 \frac{23}{5x-2} consists of all real numbers except x=25 x = \frac{2}{5} .

Therefore, the domain is expressed as x25 x \ne \frac{2}{5} .

Considering the multiple-choice options, the correct choice is:

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

Answer

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

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