integration of 2x - 3
Understand the Problem
The question is asking for the integral of the expression 2x - 3. In calculus, integration is the process of finding the antiderivative of a function, which effectively gives us the area under the curve represented by that function.
Answer
The integral of $2x - 3$ is $x^2 - 3x + C$.
Answer for screen readers
The integral of the expression $2x - 3$ is given by:
$$ x^2 - 3x + C $$
Steps to Solve
- Identify the Expression to Integrate
We need to integrate the expression $2x - 3$ with respect to $x$. This can be written as:
$$ \int (2x - 3) , dx $$
- Separate the Integral
Using the property of integrals, we can separate the integral into two parts:
$$ \int (2x - 3) , dx = \int 2x , dx - \int 3 , dx $$
- Integrate Each Part
Now we will integrate each part separately.
- For the first integral $ \int 2x , dx $, we use the power rule:
$$ \int 2x , dx = 2 \cdot \frac{x^2}{2} = x^2 $$
- For the second integral $ \int 3 , dx $, we treat 3 as a constant:
$$ \int 3 , dx = 3x $$
- Combine the Results and Add the Constant of Integration
Now combine the results from the two integrals and add the constant of integration $C$:
$$ \int (2x - 3) , dx = x^2 - 3x + C $$
The integral of the expression $2x - 3$ is given by:
$$ x^2 - 3x + C $$
More Information
The constant $C$ represents any arbitrary constant value, as the integral represents a family of functions that differ by a constant. This is a fundamental concept in calculus known as the constant of integration.
Tips
- Forgetting to add the constant of integration $C$ at the end of the integral.
- Misapplying the power rule which could lead to incorrect coefficients in the antiderivative.
