# How do you use limits to evaluate #int (x+2)dx# from [1,4]?

# int_1^4 \ (x+2) \ dx = 13.5#

By definition of an integral, then

That is

And so:

Using the standard summation formula:

we have:

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Perform the integration as if the integral were indefinite but without a constant of integration.

Subtract the expression evaluated at the lower limit from the expression evaluated at the upper limit.

Perform the integration as if the integral were indefinite but without a constant of integration:

Subtract the resulting expression evaluated at the lower limit from the expression evaluated at the upper limit:

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To evaluate the integral ∫(x+2)dx from 1 to 4 using limits, you can follow these steps:

- Integrate the function (x+2) with respect to x to find the antiderivative.
- Evaluate the antiderivative at the upper limit of integration (4) and subtract the value of the antiderivative at the lower limit of integration (1).
- This gives the definite integral of the function over the interval [1, 4].

Let's go through the steps:

- The antiderivative of (x+2) with respect to x is (1/2)x^2 + 2x + C, where C is the constant of integration.
- Evaluate the antiderivative at the upper limit (4): (1/2)(4)^2 + 2(4) = 8 + 8 = 16 Then, subtract the value of the antiderivative at the lower limit (1): (1/2)(1)^2 + 2(1) = 1/2 + 2 = 5/2 Subtract: 16 - 5/2 = 27/2
- Therefore, the definite integral of (x+2) from 1 to 4 is 27/2.

So, ∫(x+2)dx from 1 to 4 equals 27/2.

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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.

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