# How do you evaluate the definite integral #int (x^2-x+6)dx# from [1,4]?

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To evaluate the definite integral ∫(x^2 - x + 6)dx from 1 to 4, we first find the antiderivative of the function x^2 - x + 6 with respect to x, and then evaluate it at the upper limit (4) and subtract the value of the antiderivative at the lower limit (1).

The antiderivative of x^2 - x + 6 with respect to x is (1/3)x^3 - (1/2)x^2 + 6x.

So, integrating x^2 - x + 6 dx from 1 to 4:

∫(x^2 - x + 6)dx from 1 to 4 = [(1/3)x^3 - (1/2)x^2 + 6x] evaluated from 1 to 4 = [(1/3)(4)^3 - (1/2)(4)^2 + 6(4)] - [(1/3)(1)^3 - (1/2)(1)^2 + 6(1)] = [(64/3) - 8 + 24] - [(1/3) - (1/2) + 6] = [(64/3) + 16] - [(1/3) - (1/2) + 6] = (64/3 + 16) - (1/3 - 1/2 + 6) = (64/3 + 16) - (2/6 - 3/6 + 6) = (64/3 + 16) - (5/6 + 6) = (64/3 + 16) - (5/6 + 36/6) = (64/3 + 16) - (41/6) = (128/6 + 16) - (41/6) = (128 + 96)/6 - 41/6 = 224/6 - 41/6 = 183/6

Therefore, the value of the definite integral ∫(x^2 - x + 6)dx from 1 to 4 is 183/6, or 30.5.

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To evaluate the definite integral ∫(x^2 - x + 6) dx from [1, 4], we first need to find the antiderivative of the integrand. The antiderivative of x^2 - x + 6 with respect to x is (1/3)x^3 - (1/2)x^2 + 6x.

Next, we evaluate the antiderivative at the upper and lower limits of integration, which are 4 and 1, respectively.

Plugging in the upper limit (4) into the antiderivative, we get: (1/3)(4)^3 - (1/2)(4)^2 + 6(4) = (1/3)(64) - (1/2)(16) + 24 = 64/3 - 8 + 24 = 64/3 + 16.

Then, plugging in the lower limit (1) into the antiderivative, we get: (1/3)(1)^3 - (1/2)(1)^2 + 6(1) = (1/3) - (1/2) + 6 = 1/3 - 1/2 + 6 = 1/3 - 2/6 + 18/3 = 1/3 - 1/3 + 18/3 = 18/3.

Finally, we subtract the value at the lower limit from the value at the upper limit: (64/3 + 16) - (18/3) = (64/3 + 16) - 6 = 64/3 + 16 - 6 = 64/3 + 10 = (64 + 30)/3 = 94/3.

Therefore, the value of the definite integral ∫(x^2 - x + 6) dx from [1, 4] is 94/3.

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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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