How do you evaluate the definite integral #int (x+4)dx# from [-1,3]?
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To evaluate the definite integral (\int_{-1}^{3} (x+4) , dx), you can follow these steps:
- Integrate the function ((x+4)) with respect to (x) to find the antiderivative.
- Substitute the upper limit (3) into the antiderivative and subtract the result of substituting the lower limit (-1).
- Simplify the expression to get the final answer.
Here are the calculations:
-
(\int (x+4) , dx = \frac{1}{2}x^2 + 4x + C), where (C) is the constant of integration.
-
Substitute 3 into the antiderivative: (\frac{1}{2}(3)^2 + 4(3) = \frac{9}{2} + 12 = \frac{33}{2}).
Substitute -1 into the antiderivative: (\frac{1}{2}(-1)^2 + 4(-1) = \frac{1}{2} - 4 = -\frac{7}{2}).
- Subtract the results: (\frac{33}{2} - (-\frac{7}{2}) = \frac{33}{2} + \frac{7}{2} = \frac{40}{2} = 20).
Therefore, (\int_{-1}^{3} (x+4) , dx = 20).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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