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

For evaluation using the definition, please see the full explanation section below. (Skip to the end if you can use the Fundamental Theorem of Calculus.)

I am assuming that you do not yet have the Fundamental Theorem of Calculus available to evaluate this, but that you need to evaluate it from a definition .

I prefer to do this type of problem one small step at a time.

(We used summation formulas for the three sums in the previous step.)

So,

There are several ways to think about the first limit :

OR

OR

The second limit

The ideas above can be applied to get

Completing the integration

Using the Fundamental Theorem of Calculus

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To evaluate the definite integral (\int_{1}^{2} (x^2 + 1)dx), we first find the antiderivative of the integrand, which is (\frac{x^3}{3} + x).

Next, we substitute the upper limit, 2, into the antiderivative and subtract the result of substituting the lower limit, 1:

[\left(\frac{2^3}{3} + 2\right) - \left(\frac{1^3}{3} + 1\right)]

[= \left(\frac{8}{3} + 2\right) - \left(\frac{1}{3} + 1\right)]

[= \left(\frac{8}{3} + \frac{6}{3}\right) - \left(\frac{1}{3} + \frac{3}{3}\right)]

[= \frac{14}{3} - \frac{4}{3}]

[= \frac{10}{3}]

So, (\int_{1}^{2} (x^2 + 1)dx = \frac{10}{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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