Replace the improper integrand (∞ or -∞) with a placeholder variable in the improper integral. For example, in the integral from 1 to ∞ of 1 / x^2 dx, replace the ∞ integrand with to get the integral from 1 to t of 1/ x^2 dx.
Solve the definite integral from 1 to t by calculating the antiderivative of the function and evaluating it at values f(t) and f(1). In the above example, the antiderivative of 1 / (x^2) is -1/x. The integral is therefore equal to -1/t - (-1/1).
Simplify the expression in Step 2 by distributing factors, combining like terms, and reducing fractions. The expression -1/t - (-1/1) simplifies to -1/t + 1, or 1 - 1/t.
Take the limit of the simplified expression from Step 3 as the variable t goes to positive or negative infinity (whichever the sign of infinity the integrand you replaced with t in Step 1 had) and simplify the answer. In the above example, the limit as t goes to infinity of 1 - 1/t) is 1 - 0, or 1. The improper integral from 1 to ∞ of the function 1 / x^2 is therefore equal to 1.