How do you integrate #1/(x^3 +4x)# using partial fractions?
The answer is
Divide the breakdown into partial fractions.
Compare the numerators; the denominators are the same.
Consequently,
So,
Compute the second integral using substitution.
So
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Here's an additional approach to solving the issue without the
applying partial fractions.
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To integrate ( \frac{1}{x^3 + 4x} ) using partial fractions, follow these steps:

Factor the denominator:
[ x^3 + 4x = x(x^2 + 4) ] 
Express the fraction as a sum of partial fractions: [ \frac{1}{x(x^2 + 4)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 4} ]

Clear the fractions by multiplying both sides by the common denominator ( x(x^2 + 4) ): [ 1 = A(x^2 + 4) + (Bx + C)x ]

Expand and combine like terms: [ 1 = Ax^2 + 4A + Bx^2 + Cx ]

Group like terms and equate coefficients: [ Ax^2 + Bx^2 = (A + B)x^2 ]
[ 4A + Cx = 0x + 4A + C ]
From this system of equations, we can determine the values of A, B, and C:
[ A + B = 0 ]
[ 4A + C = 0 ]
 Solve for A, B, and C:
Using the first equation, ( A = B ).
Substituting this into the second equation, ( 4(B) + C = 0 )
[ 4B + C = 0 ]
[ C = 4B ]
Substituting these relationships into the first equation, ( A = B ), we get: [ A = B ] [ A = \frac{C}{4} ]

Substitute the values of A, B, and C back into the partial fraction decomposition: [ \frac{1}{x(x^2 + 4)} = \frac{B}{x} + \frac{Bx + 4B}{x^2 + 4} ]

Integrate each term separately: [ \int \frac{B}{x} , dx = B \lnx + C_1 ] [ \int \frac{Bx + 4B}{x^2 + 4} , dx = \frac{1}{2} \ln(x^2 + 4) + C_2 ]
Combine the integrals to get the final result: [ \int \frac{1}{x^3 + 4x} , dx = B \lnx + \frac{1}{2} \ln(x^2 + 4) + C ]
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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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