Evaluate the following integral: ∫ (2x(1 + sin x))/(1 + cos^2x) dx, x ∈ [-π, π]

Question

Evaluate the following integral:

\(\int\limits_{-\pi}^{\pi}\cfrac{2\text x(1+sin\text x)}{1+cos^2\text x}d\text x \)

Answer 1

  Let us assume

I = \(\int\limits_{-\pi}^{\pi}\cfrac{2\text x(1+sin\text x)}{1+cos^2\text x}d\text x \)....equation 1

By property, we know that

We know that

Sin(– x) = – sin x

Cos(– x) = cos x

We know

Thus substituting the values, we get,

By property, we know that

sin(π - x) = sin x

cos(π - x) = -cos x

Thus substituting the values

Thus substituting the given values

Substituting the values in equation 4