Indefinite integral - Examples, Exercises and Solutions

Understanding Indefinite integral

Complete explanation with examples

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

Detailed explanation

Practice Indefinite integral

Test your knowledge with 16 quizzes

Look at the following function:

\( \frac{10x-3}{5x-3} \)

What is the domain of the function?

Examples with solutions for Indefinite integral

Step-by-step solutions included
Exercise #1

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

What is the field of application of the equation?

Step-by-Step Solution

To solve this problem, we will determine the domain, or field of application, of the equation 6x+5=1 \frac{6}{x+5} = 1 .

Step-by-step solution:

  • Step 1: Identify the denominator. In the given equation, the denominator is x+5 x+5 .
  • Step 2: Determine when the denominator is zero. Solve for x x by setting x+5=0 x+5 = 0 .
  • Step 3: Solve the equation: x+5=0 x+5 = 0 gives x=5 x = -5 .
  • Step 4: Exclude this value from the domain. The domain is all real numbers except x=5 x = -5 .

Therefore, the field of application of the equation is all real numbers except where x=5 x = -5 .

Thus, the domain is x5 x \neq -5 .

Answer:

x5 x\operatorname{\ne}-5

Video Solution
Exercise #2

x+y:32x+6=4 \frac{x+y:3}{2x+6}=4

What is the field of application of the equation?

Step-by-Step Solution

To solve this problem, we'll follow these steps to find the domain:

  • Step 1: Recognize that the expression x+y:32x+6=4\frac{x+y:3}{2x+6}=4 involves a fraction. The denominator 2x+62x + 6 must not be zero, as division by zero is undefined.
  • Step 2: Set the denominator equal to zero and solve for xx to find the values that must be excluded: 2x+6=02x + 6 = 0.
  • Step 3: Solve 2x+6=02x + 6 = 0:
    • 2x+6=02x + 6 = 0
    • 2x=62x = -6
    • x=3x = -3
  • Step 4: Conclude that the domain of the function excludes x=3x = -3, meaning x3x \neq -3.

Thus, the domain of the given expression is all real numbers except x=3x = -3. This translates to:

x3 x\operatorname{\ne}-3

Answer:

x3 x\operatorname{\ne}-3

Video Solution
Exercise #3

3x:4y+6=6 \frac{3x:4}{y+6}=6

What is the field of application of the equation?

Step-by-Step Solution

To determine the field of application of the equation 3x:4y+6=6\frac{3x:4}{y+6}=6, we must identify values of yy for which the equation is defined.

  • The denominator of the given expression is y+6y + 6. In order for the expression to be defined, the denominator cannot be zero.
  • This leads us to solve the equation y+6=0y + 6 = 0.
  • Solving y+6=0y + 6 = 0 gives us y=6y = -6.
  • This means y=6y = -6 would make the denominator zero, thus the expression would be undefined for this value.

Therefore, the field of application, or the domain of the equation, is all real numbers except y=6y = -6.

We must conclude that y6 y \neq -6 .

Comparing with the provided choices, the correct answer is choice 3: y6 y \neq -6 .

Answer:

y6 y\operatorname{\ne}-6

Video Solution
Exercise #4

Given the following function:

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

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

Step-by-Step Solution

To determine the domain of the function 5x2x \frac{5-x}{2-x} , we need to identify and exclude any values of x x that make the function undefined. This occurs when the denominator equals zero.

  • Step 1: Set the denominator equal to zero:
    2x=0 2-x = 0
  • Step 2: Solve for x x :
    Adding x x to both sides gives 2=x 2 = x . Hence, x=2 x = 2 .

This means that the function is undefined when x=2 x = 2 . Therefore, the domain of the function consists of all real numbers except x=2 x = 2 .

Thus, the domain is: x2 x \ne 2 .

The correct answer choice is:

Yes, x2 x\ne2

Answer:

Yes, x2 x\ne2

Video Solution
Exercise #5

Given the following function:

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

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

Step-by-Step Solution

To determine the domain of the function 49+2xx+4 \frac{49 + 2x}{x + 4} , we need to focus on avoiding division by zero, which occurs when the denominator is zero.

Let's identify the denominator of the function:

  • The denominator is x+4 x + 4 .

Next, we set the denominator equal to zero and solve for x x :

  • x+4=0 x + 4 = 0
  • Subtract 4 from both sides: x=4 x = -4

This calculation shows that the function is undefined when x=4 x = -4 . Thus, the domain of the function is all real numbers except x=4 x = -4 .

Therefore, the domain of the function is x4 x \neq -4 .

In terms of the provided choices, this corresponds to choice 4:

Yes, x4 x \ne -4

Answer:

Yes, x4 x\ne-4

Video Solution

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