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

[vande060]

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

 

[rock.freak667]

Try completing the square for x^2 + x+ 1 and then make an appropriate substitution.

 

[vande060]

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

 

[tiny-tim]

hi vande060!

√((u)^2 + 3/4)

u = (√3/2)secϑ

oooh … learn all your trigonometric identities …

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

 

[vande060]

hi vande060! oooh … learn all your trigonometric identities …

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

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