Integration of x - 1
Understand the Problem
The question is asking for the integral of the function x - 1. This involves performing a mathematical operation to find the antiderivative of the function.
Answer
$\frac{x^2}{2} - x + C$
Answer for screen readers
The integral of x - 1 is (x^2)/2 - x + C, where C is the constant of integration.
Steps to Solve
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Integrate each term
We integrate each term of the function separately:
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For $x$: The integral of $x$ is $\frac{x^2}{2}$
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For $-1$: The integral of a constant is the constant times $x$, so $-x$
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Combine terms
We combine the integrated terms:
$$\frac{x^2}{2} - x$$
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Add constant
We add the constant of integration $C$:
$$\frac{x^2}{2} - x + C$$
This constant represents the fact that there are infinitely many antiderivatives that differ by a constant.
The integral of x - 1 is (x^2)/2 - x + C, where C is the constant of integration.
More Information
The integral of a function represents the area under the curve of that function. In this case, the integral of x - 1 gives us a quadratic function, which can be used to calculate the area between the line y = x - 1 and the x-axis over any given interval.
Tips
A common mistake is forgetting to add the constant of integration C. Remember that when you differentiate any constant, it becomes zero, so we need to account for this possibility in integration by adding an arbitrary constant.
