Evaluate the following Integrals:

Let

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

Put,

(Differentiating both sides)

So, the new limits are ∞ and 2.

Substituting this in the original integral,

Recall,