# How do you evaluate the indefinite integral #int (x^3+3x+1)dx#?

# int \ x^3+3x+1 \ dx = x^4/4+(3x^2)/2+x + C #

When integrating, we apply the power rule:

Thus, we have:

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To evaluate the indefinite integral (\int (x^3 + 3x + 1) , dx), you can use the rules of integration.

The integral of (x^n) with respect to (x) is (\frac{x^{n+1}}{n+1} + C), where (C) is the constant of integration.

So, applying this rule to each term of the given function:

[ \begin{aligned} \int (x^3 + 3x + 1) , dx &= \int x^3 , dx + \int 3x , dx + \int 1 , dx \ &= \frac{x^4}{4} + \frac{3x^2}{2} + x + C, \end{aligned} ]

where (C) is the constant of integration.

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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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