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Tricky Question...

Hi could someone help me with the following question plz:

Find the following indefinite integral -

∫ dx / x(x²+x+1)

Thanks.

Reply 1

Jonny W

The identity

1 / x(x^2 + x + 1)
= A/x + (Bx + C)/(x^2 + x + 1)

holds if

1 = A(x^2 + x + 1) + (Bx + C)x

Compare coefficients of x^2: A + B = 0.
Compare coefficients of x: A + C = 0.
Compare constants: A = 1.

So

1 / x(x^2 + x + 1)
= 1/x - (x + 1) / (x^2 + x + 1)
= 1/x - (1/2)(2x + 1) / (x^2 + x + 1) - (1/2) / (x^2 + x + 1)
= 1/x - (1/2)(2x + 1) / (x^2 + x + 1) - (1/2) / [(x + 1/2)^2 + 3/4]

(int) 1 / x(x^2 + x + 1) dx
= ln(x) - (1/2)ln(x^2 + x + 1) - (1/2)(2/sqrt(3))arctan[(2/sqrt(3))(x + 1/2)] + c
= ln(x) - (1/2)ln(x^2 + x + 1) - (1/sqrt(3))arctan[(2x + 1)/sqrt(3)] + c