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

Question

Evaluate the following Integral:

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

Answer 1

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

As we have the trigonometric identity 1 + tan²θ = sec²θ, to evaluate this integral we use x = tan θ

⇒ dx = sec²θ dθ (Differentiating both sides)

When x = 0, tan θ = 0 ⇒ θ = 0

When x = 1, tan θ = 1 ⇒ θ = \(\cfrac{\pi}4\)

So, the new limits are 0 and \(\cfrac{\pi}4\).

Substituting this in the original integral,