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

Answer 1

#arctan((x+1/2)/(sqrt(3)/2)) /(sqrt(3)/2)#

#x^2+x+1# has two complex conjugate roots. Let us generalize solving for
#1/((x+a)^2+b^2) = A/(x+a+jb)+B/(x+a-jb)#

formulating and establishing coefficients

#{(a (A + B) - j b(A- B) = 1), (A + B=0):}#
Solving for #{A,B}# we get
#A = j/(2b),B=-j/(2b)#
#int dx/((x+a)^2+b^2) = j int dx/(2 b (x + a + j b)) -j int dx/(2 b (x + a -j b))#

then

#int dx/((x+a)^2+b^2) = log_e((j(x+a)+b)/(j(x+a)-b))^{1/(j2b)}#

but

#j(x+a)+b = (sqrt((x+a)+b^2))e^{j phi}# #j(x+a)-b = (sqrt((x+a)+b^2))e^{-j phi}#
with #phi = arctan((x+a)/a)#

then

#int dx/((x+a)^2+b^2) = log_e(e^{2j phi))^{1/(j2b)} = log_e(e^{phi/b}) = phi/b#

Lastly, we have

#int dx/((x+a)^2+b^2) = arctan((x+a)/b)/b#
For #a = 1/2, b = sqrt(3/4)#
#int dx/((x+1/2)^2+(sqrt(3)/2)^2) =arctan((x+1/2)/(sqrt(3)/2)) /(sqrt(3)/2)#
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Answer 2

To integrate ( \int \frac{1}{x^2+x+1} ) using partial fractions:

  1. Factor the denominator ( x^2 + x + 1 ) if possible. If it's irreducible, proceed to step 2.
  2. Express ( \frac{1}{x^2+x+1} ) as ( \frac{A}{x + \alpha} + \frac{B}{x + \beta} ) where ( \alpha ) and ( \beta ) are the roots of the quadratic equation ( x^2 + x + 1 = 0 ).
  3. Solve for ( A ) and ( B ) by multiplying both sides by the denominator ( x^2+x+1 ) and simplifying.
  4. After finding ( A ) and ( B ), rewrite ( \frac{1}{x^2+x+1} ) as ( \frac{A}{x + \alpha} + \frac{B}{x + \beta} ).
  5. Integrate each fraction separately.
  6. Finally, put the integral back together to get the solution.

This method allows you to integrate the given function using partial fractions.

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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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