Integral of 1/x² from −1 to 1

The integral diverges.

The integrand 1/x² is not defined at x = 0, which lies inside [−1, 1], and it is unbounded there. So this is an improper integral and must be split at 0:

∫(−1)^1 dx/x² = ∫(−1)^0 dx/x² + ∫₀¹ dx/x²

Consider the second part:

∫₀¹ dx/x² = lim_(ε→0⁺) ∫_ε¹ x⁻² dx
          = lim [−1/x]_ε¹
          = lim (−1 + 1/ε)
          = +∞

Since one part diverges (the other part also diverges to +∞ by symmetry), the integral does not exist as a finite number.

Warning: applying the fundamental theorem blindly gives [−1/x]_(−1)^1 = −1 − 1 = −2, which is wrong. The integrand is positive, so a negative answer is impossible, and the theorem requires the integrand to be continuous on the whole interval.

Answer: the integral diverges (equals +∞).