Rational Splits and Infinite Limits
Two closing techniques: break rational integrands into simple pieces, and push integrals out to infinity.
Partial Fractions
A rational function with a factored denominator can be written as a sum of simpler fractions, one per factor. Solve for the unknown numerators by clearing denominators and matching coefficients. Each piece integrates with a logarithm or an arctangent, which is the whole point of the split.
\frac{1}{x^2-1} = \frac{1/2}{x-1} - \frac{1/2}{x+1}Improper Limits
An integral is improper when an endpoint is infinite or the integrand blows up inside the interval. Replace the bad limit with a variable t, integrate normally, then take a limit. If the limit settles on a finite number the integral converges; otherwise it diverges and area is not assigned.
\int_a^{\infty} f(x)\,dx = \lim_{t\to\infty}\int_a^t f(x)\,dxA Worked Pair
Split one over (x times (x plus 1)) as one over x minus one over (x plus 1), and the integral is the natural log of x over (x plus 1) plus C. For the infinite side, the integral of one over x squared from 1 to infinity converges to 1, while one over x diverges. The exponent, not the constant, decides the outcome.
\int_1^{\infty}\frac{dx}{x^2} = \lim_{t\to\infty}\left(1-\frac{1}{t}\right) = 1Flashcard
Partial fraction form for two linear factors
Check your understanding
Does the integral of one over x from 1 to infinity converge, and why?