# Evaluate the following Integral: ∫(1 - x^2)/(x^4+x^2+1) dx, x ∈ [0, 1]

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Evaluate the following Integral:

$\int\limits_0^1\cfrac{1-\text x^2}{\text x^4+\text x^2+1}d\text x$

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Let I = $\int\limits_0^1\cfrac{1-\text x^2}{\text x^4+\text x^2+1}d\text x$

In the denominator, we have x4 + x2 + 1

Note that we can write x4 + x2 + 1 = (x4 + 2x2 + 1) – x2

We have x4 + 2x2 + 1 = (1 + x2)2

⇒ x4 + x2 + 1 = (1 + x2)2 – x2

So, using this, we can write our integral as

Dividing numerator and denominator with x2, we have

(Differentiating both sides)

So, the new limits are ∞ and 2.

Substituting this in the original integral,