# How do you evaluate the definite integral #int (2x^3)dx# from [1,3]?

We will start by thinking about the integral as I with the limits:

The constant will be removed from the multiplication with the variable as it is:

We are aware of the integration power rule:

Using the integral and the power rule

We can now set the integration's boundaries because we are aware of the following rule:

Higher and Lower Boundaries

Hence:

Hence:

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To evaluate the definite integral of ( \int_{1}^{3} 2x^3 , dx ), we first find the antiderivative of the integrand. In this case, the antiderivative of ( 2x^3 ) is ( \frac{1}{2}x^4 ). Then, we apply the Fundamental Theorem of Calculus, which states that if ( F(x) ) is an antiderivative of ( f(x) ) on an interval ([a, b]), then (\int_{a}^{b} f(x) , dx = F(b) - F(a)).

So, ( \int_{1}^{3} 2x^3 , dx = \left[ \frac{1}{2}x^4 \right]_{1}^{3} = \frac{1}{2}(3^4) - \frac{1}{2}(1^4) = \frac{81}{2} - \frac{1}{2} = \frac{80}{2} = 40 ).

Therefore, the value of the definite integral is 40.

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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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