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

Question

Evaluate the following integral:

\(\int\limits_{0}^{\infty}\cfrac{\text x}{(1+\text x)(1+\text x^2)}d\text x \)

Answer 1

Let us assume

I = \(\int\limits_{0}^{\infty}\cfrac{\text x}{(1+\text x)(1+\text x^2)}d\text x \)

Adding – 1 and + 1

Let I₁ = \(\int\limits_{0}^{\infty}\cfrac{1}{(1+\text x^2)}d\text x \)

I₂ = \(\int\limits_{0}^{\infty}\cfrac{\text x}{(1+\text x)(1+\text x^2)}d\text x \)

Thus I = I₁ – I₂ …….equation 1

Solving for I₁

Let

a + b = 0; a + c = 1; b + c = 0

solving we get

a = c = 1/2 

b = – 1/2

substituting the values in equation 3

Solving:

Thus

Substituting values equation 2 and equation 4 in equation 1

Thus