integral of 2x - 3
Understand the Problem
The question is asking for the integral of the function 2x - 3. To solve this, we will apply the fundamental rules of integration to find the antiderivative of the given expression.
Answer
The integral of \( 2x - 3 \) is \( x^{2} - 3x + C \).
Answer for screen readers
The integral of the function ( 2x - 3 ) is given by:
$$ \int (2x - 3) , dx = x^{2} - 3x + C $$
Steps to Solve
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Identifying the Function to Integrate
We need to integrate the function ( f(x) = 2x - 3 ). -
Applying the Integral
To find the integral, we apply the formula for the integral of a polynomial. The integral of ( ax^n ) is ( \frac{a}{n+1} x^{n+1} + C ), where ( C ) is the constant of integration.
Here, we can split the integral:
$$ \int (2x - 3) , dx = \int 2x , dx - \int 3 , dx $$
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Calculating the Integral of Each Term
Now we can work on each integral separately.
For ( \int 2x , dx ):
$$ \int 2x , dx = 2 \cdot \frac{1}{2} x^{2} = x^{2} $$
For ( \int 3 , dx ):
$$ \int 3 , dx = 3x $$
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Combining the Results
Now we combine the results obtained from the two integrals and include the constant of integration ( C ):
$$ \int (2x - 3) , dx = x^{2} - 3x + C $$
The integral of the function ( 2x - 3 ) is given by:
$$ \int (2x - 3) , dx = x^{2} - 3x + C $$
More Information
The process of finding the integral is essentially finding the antiderivative of the function. This is a fundamental concept in calculus that allows us to compute the area under a curve defined by a function.
Tips
- Forgetting to include the constant of integration ( C ): Always remember to add the constant after integrating.
- Mistakes in applying the power rule for integration: Ensure to apply ( \frac{a}{n+1} x^{n+1} ) accurately.
