Evaluate ∫ dx/x³ from ∞ to α

Note: the limits are printed as ∞ (lower) and α (upper). We evaluate exactly as written, and also give the usual version ∫_α^∞.

∫ dx/x³ = ∫ x⁻³ dx = −1/(2x²).

As written (taking α > 0):

∫_∞^α dx/x³ = [−1/(2x²)]_∞^α
            = −1/(2α²) − lim_(x→∞)(−1/(2x²))
            = −1/(2α²) − 0
            = −1/(2α²)

Reversing the limits changes the sign, so this equals −∫_α^∞ dx/x³.

The usual version ∫_α^∞ dx/x³ = [−1/(2x²)]_α^∞ = 0 + 1/(2α²) = 1/(2α²), which converges (for α > 0).

Answer: ∫_∞^α dx/x³ = −1/(2α²) (and ∫_α^∞ dx/x³ = 1/(2α²)).