How do you integrate #(4x^3+2x^2+1)/(4x^3-x)# using partial fractions?

Answer 1

The answer is #=x-ln(|x|)+1/2ln(|2x+1|)+ln(|2x-1|)+C#

As the degree of the numerator is equal to the degree of the denominator, perform a polynomial long division

#(4x^3+2x^2+1)/(4x^3-x)=1+(2x^2+x+1)/(4x^3-x)#
#=1+(2x^2+x+1)/(x(4x^2-1))#
#=1+(2x^2+x+1)/(x(2x+1)(2x-1))#

Perform the partial fraction decomposition

#(2x^2+x+1)/(x(2x+1)(2x-1))=A/(x)+B/(2x+1)+C/(2x-1)#
#=(A(2x+1)(2x-1)+B(x)(2x-1)+C(x)(2x+1))/(x(2x+1)(2x-1))#

The denominators are the same, compare the numerators

#2x^2+x+1=A(2x+1)(2x-1)+B(x)(2x-1)+C(x)(2x+1)#
Let #x=0#, #=>#, #1=-A#, #=>#, #A=-1#
Let #x=-1/2#, #=>#, #1=B#
Let #x=1/2#, #=>#, #C=2#

Therefore,

#(4x^3+2x^2+1)/(4x^3-x)=1-1/(x)+1/(2x+1)+2/(2x-1)#
#int((4x^3+2x^2+1)dx)/(4x^3-x)=int1dx-int(1dx)/(x)+int(1dx)/(2x+1)+int(2dx)/(2x-1)#
#=x-ln(|x|)+1/2ln(|2x+1|)+ln(|2x-1|)+C#
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Answer 2

To integrate (4x^3 + 2x^2 + 1) / (4x^3 - x) using partial fractions, you first need to factor the denominator and then express the rational function as a sum of simpler fractions. The denominator (4x^3 - x) can be factored as x(4x^2 - 1), which further simplifies to x(2x + 1)(2x - 1). Therefore, the partial fraction decomposition will have the form:

(4x^3 + 2x^2 + 1) / (4x^3 - x) = A/x + B/(2x + 1) + C/(2x - 1)

To find the constants A, B, and C, you typically clear the fractions by multiplying both sides of the equation by the denominator of the original expression. Then, you solve for the constants by equating coefficients of like terms on both sides of the equation.

After finding the values of A, B, and C, you integrate each term separately, which will give you the final result.

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Answer from HIX Tutor

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.

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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