# Indefinite integral

🏆Practice domain of a function

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.

## Test yourself on domain of a function!

Given the following function:

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

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

• 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.

## Examples and exercises with solutions of indefinite integral

### Exercise #1

Given the following function:

$\frac{9x}{4}$

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

### Step-by-Step Solution

Since the function's denominator equals 4, the domain of the function is all real numbers, meaning all X.

No, the entire domain

### Exercise #2

Given the following function:

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

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

### 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:

$(2x-2)^2\ne0$

$2x\ne2$

$x\ne1$

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

Yes, $x\ne1$

### Exercise #3

Given the following function:

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

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

### 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.

No, the entire domain

### Exercise #4

Given the following function:

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

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

### Video Solution

Yes, $x\ne2$

### Exercise #5

Given the following function:

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

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

### Video Solution

Yes, $x\ne-4$