# How do you evaluate the integral #int (x^3-4)/(x+1)#?

Integrate term by term:

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To evaluate the integral ∫(x^3 - 4)/(x + 1), perform polynomial long division to rewrite the integrand as x^2 - x + 1 - 3/(x + 1). Then, integrate each term separately. The integral of x^2 - x + 1 with respect to x is (1/3)x^3 - (1/2)x^2 + x, and the integral of -3/(x + 1) can be found using the natural logarithm. Therefore, the value of the integral is (1/3)x^3 - (1/2)x^2 + x - 3ln|x + 1| + C, where C is the constant of integration.

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To evaluate the integral ( \int \frac{x^3 - 4}{x + 1} ), you can use polynomial long division or partial fraction decomposition. Here, I'll demonstrate how to use partial fraction decomposition:

- First, factor the numerator ( x^3 - 4 ) if possible. Since it's a difference of cubes, you can factor it as ( (x - \sqrt[3]{4})(x^2 + \sqrt[3]{4}x + \sqrt[3]{16}) ).
- Write the fraction as a sum of simpler fractions by decomposing it into partial fractions.
- The degree of the numerator is 3, and the degree of the denominator is 1, so the decomposition will include both linear and constant terms.
- Assume the decomposition is of the form ( \frac{x^3 - 4}{x + 1} = Ax^2 + Bx + C ).
- Multiply both sides by ( x + 1 ) to clear the denominator.
- Equate coefficients of corresponding powers of ( x ) on both sides.
- Solve the resulting system of equations for ( A ), ( B ), and ( C ).
- Once you find ( A ), ( B ), and ( C ), integrate each term separately.
- The integral of the original function is the sum of the integrals of the decomposed terms.

By following these steps, you can evaluate the integral of ( \frac{x^3 - 4}{x + 1} ).

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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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