The integral 24 π ∫ 0 2 2 - x 2 dx 2 + x 2 4 + x 4 is equal to ______.

Step-by-step solution

Let, I = 24 π ∫ 0 2 2 - x 2 2 + x 2 4 + x 4 d x = 24 π ∫ 0 2 2 - x 2 x 2 2 x + x 4 x 2 + x 2 d x = 24 π ∫ 0 2 2 x 2 - 1 2 x + x 2 x + x 2 - 4 d x Now let 2 x + x = t ,   - 2 x 2 + 1 d x = d t I = - 24 π ∫ ∞ 2 2 d t t t 2 - 4 = - 12 π ∫ ∞ 2 2 2 t d t t 2 t 2 - 4 Let t 2 - 4 = z 2 ,   2 t d t = 2 z d z I = - 12 π ∫ ∞ 2 2 z d z z z 2 + 4 = - 24 π ∫ ∞ 2 d z z 2 + 4 = - 24 π 1 2 tan - 1 z 2 ∞ 2 = - 2