Indefinite integral

An integral can be defined for all values (that is, for all $X$ ). 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 $X$ ), since if we place certain $X$ or a certain range of values of $X$ we will receive an expression considered "invalid" in mathematics. The values of $X$ for which integration is undefined cause the discontinuity of a function.

Detailed explanation

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Test yourself on domain of a function!

Look at the following function:

$\frac{5}{x}$

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

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An example of this is a function with a fraction with values $X$ in the denominator.
For example $1\over x$
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 $X$ that could cause the denominator to be zero.

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 $X$ that cause the expression under the square root to be negative. $f(x)=\sqrt{x^2-x-5}$

Example of a negative square root function

If you are interested in this article, you might also be interested in the following articles:

Graphical representation of a function

Algebraic representation of a function

Function notation

Domain of a function

Numeric value assignment in a function

Variation of a function

Increasing function

Decreasing function

Constant function

Functions for seventh grade

Intervals of increase and decrease of a function

On the Tutorela website, you will find a variety of articles with interesting explanations about mathematics

Examples and exercises with solutions of indefinite integral

Exercise #1

Look at the following function:

$\frac{5}{x}$

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, $x\ne0$

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, $x\ne0$

Exercise #2

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

$\frac{9x}{4}$

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 #3

Look at the following function:

$\frac{10x}{\frac{1}{2}}$

What is the domain of the function?

Video Solution

Step-by-Step Solution

To solve this problem, follow these steps:

Step 1: Simplify the expression.

Given the function:

$f(x) = \frac{10x}{\frac{1}{2}}$

We can simplify this expression by multiplying by the reciprocal of the denominator:

$f(x) = 10x \times 2 = 20x$

Step 2: Determine the domain.

Since $f(x) = 20x$ is a linear function, it is defined for all real numbers. There are no restrictions on $x$ since no division by zero or any undefined operations are present.

Conclusion: The domain of the function is all real numbers. This corresponds to choice :

All real numbers

Therefore, the domain of the function is all real numbers.

Answer

All real numbers

Exercise #4

$22(\frac{2}{x}-1)=30$

What is the domain of the equation above?

Video Solution

Step-by-Step Solution

To find the domain of the given function $22\left(\frac{2}{x} - 1\right) = 30$ , follow these steps:

Identify critical terms: The term $\frac{2}{x}$ is undefined when $x = 0$ because division by zero is undefined.
We need to exclude $x = 0$ from the domain to ensure the function remains defined.
The correct domain for the equation is all real numbers except $x = 0$ .

Thus, the domain of the equation is $x \neq 0$ .

Therefore, the solution to the problem is $x \neq 0$ .

Answer

x≠0

Exercise #5

$2x+\frac{6}{x}=18$

What is the domain of the above equation?

Video Solution

Step-by-Step Solution

To solve this problem and find the domain for the expression $2x + \frac{6}{x}$ , we apply the following steps:

Step 1: Identify when the fraction $\frac{6}{x}$ is undefined. This occurs when the denominator $x$ equals zero.
Step 2: To find the restriction, set the denominator equal to zero: $x = 0$ .
Step 3: Solve for $x$ to find the values excluded from the domain. Here, $x \neq 0$ .

Since $\frac{6}{x}$ is undefined for $x = 0$ , the value $x = 0$ must be excluded from the domain.
Hence, the domain of the equation is all real numbers except zero.