How do you use partial fractions to find the integral #int (x^3-x+3)/(x^2+x-2)dx#?

Answer 1

THe answer is #=x^2/2-x+ln(∣x-1∣)+ln(∣x+2∣)+C#

The denominator is

#x^2+x-2=(x-1)(x+2)#

But before, let's do a long division

#color(white)(aaaa)##x^3-x+3##color(white)(aaaa)##∣##x^2+x-2#
#color(white)(aaaa)##x^3+x^2-2x##color(white)(aa)##∣##x-1#
#color(white)(aaaa)##0-x^2+x+3#
#color(white)(aaaaaa)##-x^2-x+2#
#color(white)(aaaaaaaa)##0+2x+1#

So,

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

Now we do the decomposition in partial fractions

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

so,

#2x+1=A(x+2)+B(x-1))#
Let #x=1#, #=>#, #3=3A#, #=>#, #A=1#
Let #x=-2#, #=>#, #-3=-3B#, #=>#, #B=1#

So,

#(x^3-x+3)/(x^2+x-2)=(x-1)+1/(x-1)+1/(x+2)#
#int((x^3-x+3)dx)/(x^2+x-2)=int(x-1)dx+intdx/(x-1)+intdx/(x+2)#
#=x^2/2-x+ln(∣x-1∣)+ln(∣x+2∣)+C#
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Answer 2

To find the integral ( \int \frac{{x^3 - x + 3}}{{x^2 + x - 2}} , dx ) using partial fractions, first factor the denominator as ( (x - 1)(x + 2) ). Then, express the fraction as a sum of partial fractions:

[ \frac{{x^3 - x + 3}}{{x^2 + x - 2}} = \frac{{A}}{{x - 1}} + \frac{{B}}{{x + 2}} ]

Next, multiply both sides by the denominator ( (x - 1)(x + 2) ) to clear the fractions. This gives:

[ x^3 - x + 3 = A(x + 2) + B(x - 1) ]

Now, solve for ( A ) and ( B ) by equating coefficients of corresponding powers of ( x ). Once you find ( A ) and ( B ), integrate each term separately to find the integral.

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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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