# How do you find the definite integral for: #((2x^(2) +3)dx# for the intervals #[4, 6]#?

The antiderivative can be found using the rule:

So, we can express the definite integral as:

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To find the definite integral of ( (2x^2 + 3) ) over the interval ([4, 6]), follow these steps:

- Find the antiderivative of (2x^2 + 3) with respect to (x).
- Evaluate the antiderivative at the upper limit of integration (6).
- Evaluate the antiderivative at the lower limit of integration (4).
- Subtract the result from step 3 from the result from step 2.

Let's proceed with these steps:

- The antiderivative of (2x^2 + 3) with respect to (x) is (\frac{2}{3}x^3 + 3x + C), where (C) is the constant of integration.
- Evaluating the antiderivative at (x = 6), we get (\left(\frac{2}{3}(6)^3 + 3(6)\right) = 144 + 18 = 162).
- Evaluating the antiderivative at (x = 4), we get (\left(\frac{2}{3}(4)^3 + 3(4)\right) = 85.\overline{3} + 12 = 97.\overline{3}).
- Subtracting the result from step 3 from the result from step 2, we get (162 - 97.\overline{3} = 64.\overline{6}).

Therefore, the definite integral of ( (2x^2 + 3) ) over the interval ([4, 6]) is (64.\overline{6}).

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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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