How do you use partial fraction decomposition to decompose the fraction to integrate #(x^2x6)/(x^3 +3x)#?
First we need to factor the denominator into quadratic and linear polynomials that are irreducible using Real coefficients.
Clear the denominators or get a common denominator on the left to see that we need:
Setting the coefficients equal to each other, we need to solve:
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To use partial fraction decomposition to decompose the fraction ( \frac{x^2  x  6}{x^3 + 3x} ), follow these steps:
 Factor the denominator ( x^3 + 3x ) if possible.
 Write the given fraction in the form of partial fractions with undetermined coefficients.
 Clear the fractions by multiplying both sides of the equation by the common denominator.
 Equate coefficients of like terms on both sides of the equation.
 Solve for the unknown coefficients.
 Integrate the decomposed fractions individually.
Here are the steps applied to the given fraction:

Factor the denominator ( x^3 + 3x ): [ x^3 + 3x = x(x^2 + 3) = x(x + \sqrt{3}i)(x  \sqrt{3}i) ]

Write the given fraction in the form of partial fractions: [ \frac{x^2  x  6}{x(x^2 + 3)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 3} ]

Clear the fractions by multiplying both sides by the common denominator ( x(x^2 + 3) ): [ x^2  x  6 = A(x^2 + 3) + (Bx + C)x ]

Equate coefficients of like terms: [ x^2  x  6 = Ax^2 + 3A + Bx^2 + Cx ]

Solve for the unknown coefficients:
 Equate coefficients of ( x^2 ): ( A + B = 1 )
 Equate coefficients of ( x ): ( A + C = 1 )
 Equate constant terms: ( 3A = 6 )
Solving these equations: [ A = 2 ] [ B = 3 ] [ C = 1 ]
 Rewrite the original fraction using the decomposed form: [ \frac{x^2  x  6}{x(x^2 + 3)} = \frac{2}{x} + \frac{3x  1}{x^2 + 3} ]
Now, you can integrate each term separately.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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