Trig Substitution for ∫ x/(x^2 + x+ 1)dx: Simplifying Complex Integrals

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In summary, the problem is to evaluate the integral of x/(x^2 + x+ 1)dx and the suggested approach is to complete the square for x^2 + x+ 1 and make an appropriate substitution. After completing the square, the suggested substitution is u = (x+1/2) and du = dx. By using the trigonometric identity sec^2ϑ = tan^2ϑ + 1, the integral can be rewritten in terms of trigonometric functions and solved from there.
  • #1
vande060
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Homework Statement



∫ x/(x^2 + x+ 1)dx

Homework Equations


The Attempt at a Solution



∫ x/(x^2 + x+ 1)dx

not really sure where to start on this one, i feel like i should factor the denominator in such a way that i have an expression whose derivative is some constant times x. help please
 
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  • #2
Try completing the square for x^2 + x+ 1 and then make an appropriate substitution.
 
  • #3
rock.freak667 said:
Try completing the square for x^2 + x+ 1 and then make an appropriate substitution.

∫ x/√(x^2 + x + 1 -3/4 + 3/4)

√((x + 1/2)^2 + 3/4)

u = (x+1/2)
u = dx

√((u)^2 + 3/4)

u = (√3/2)tanϑ du = √3/2sec^2ϑdϑ

√(u^2 + 3/4) = √3/2secϑ- pi/2 < ϑ < pi/2

then substituting things back in

((√3/2)tanϑ - 1/2)/ √3/2secϑ)* √3/2sec^2ϑdϑ

im weary of that square root in the numerator
 
Last edited:
  • #4
hi vande060! :smile:
vande060 said:
√((u)^2 + 3/4)

u = (√3/2)secϑ

oooh :cry: … learn all your trigonometric identities …

sec2 = tan2 + 1, not t'other way round :redface:
 
  • #5
tiny-tim said:
hi vande060! :smile:oooh :cry: … learn all your trigonometric identities …

sec2 = tan2 + 1, not t'other way round :redface:

fixed, if this is correct so far, i can finish it out myself
 

Related to Trig Substitution for ∫ x/(x^2 + x+ 1)dx: Simplifying Complex Integrals

1. What is trig substitution?

Trig substitution is a technique used in calculus to simplify integrals involving expressions with trigonometric functions. It involves replacing trigonometric expressions with simpler expressions involving only a single variable.

2. When is trig substitution used?

Trig substitution is typically used when dealing with integrals that involve the square root of a quadratic expression or when working with integrals that contain expressions of the form (a^2 - x^2)^(1/2).

3. How does trig substitution work?

To use trig substitution, we first identify the form of the integral and then select an appropriate trigonometric substitution to replace the variable. We then use trigonometric identities to simplify the integral and solve for the original variable.

4. What are the common trig substitutions used?

The most commonly used trig substitutions are:
1) x = a sinθ
2) x = a tanθ
3) x = a secθ
4) x = a cosθ
Where a is a constant and θ is a new variable.

5. Are there any specific rules to follow when using trig substitution?

Yes, there are a few rules to follow when using trig substitution:
1) Always make sure to substitute the variable back in at the end
2) Be careful with the signs of the trig functions
3) Use trig identities to simplify the integral
4) Watch out for the limits of integration and make appropriate changes
5) Check your final answer for correctness.

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