How do you use partial fraction decomposition to decompose the fraction to integrate #(x^5 + 1)/(x^6 - x^4)#?

Answer 1

#(x^5 + 1)/(x^6 - x^4) = -1/x^2-1/x^4+1/(x-1)#

Factor the denominator:

#x^6-x^4 - x^4(x^2-1) = x^4(x+1)(x-1)#
Perhaps we know that #x^5+1# can be factored, or perhaps we notice that #-1# is a zero of the numerator, so #x-1# is a factor.

In any case, we can write:

#(x^5 + 1)/(x^6 - x^4) = ((x+1)(x^4-x^3+x^2-x+1))/(x^4(x+1)(x-1))#

So

#(x^5 + 1)/(x^6 - x^4) = (x^4-x^3+x^2-x+1)/(x^4(x-1))#
#A/x+B/x^2+C/x^3+D/x^4+E/(x-1) = (x^4-x^3+x^2-x+1)/(x^4(x-1))#

Leads to:

#(Ax^4-Ax^3+Bx^3-Bx^2+Cx^2-Cx+Dx-D+Ex^4)/(x^4(x-1))#
# = (x^4-x^3+x^2-x+1)/(x^4(x-1))#
So #A+E = 1# #-A+B=-1# #-B+C = 1# #-C+D=-1# #-D = 1#
So #D=-1# and,working back through the list:
#C =0" "#, #B = -1" "#, #A = 0" "#, and #" "E = 1#.

But what if? What if we didn't notice that the fraction can be reduced?

It's OK, just a bit longer.

We would get:

#(x^5 + 1)/(x^6 - x^4) = (x^5+1)/(x^4(x+1)(x-1))#
#A/x+B/x^2+C/x^3+D/x^4+E/(x-1) +F/(x+1) = (x^5+1)/(x^4(x+1)(x-1))#

In this case we get one more variable (and one more equation).

When we solve, we'll get the same #A, B, C, D, "and " E# and we will get #F=0#
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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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