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

Question

Evaluate the following Integral:

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

Answer 1

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,