What if both bounds are infinite?

Split the integral at any convenient finite point $c$, typically $0$. Evaluate $\int_{-\infty}^{c} f(x)dx + \int_{c}^{\infty} f(x)dx$ using two separate limits. Both parts must converge independently for the entire integral to converge.

What if the discontinuity is in the middle of the interval?

Split the integral into two separate improper integrals at the point of discontinuity. The original integral converges only if both of the newly formed integrals converge.

How does the p-test apply to improper integrals?

For integrals of the form $\int_{1}^{\infty} \frac{1}{x^p} dx$, the integral converges if $p > 1$ and diverges if $p \leq 1$.

How to evaluate an improper integral

An improper integral evaluates the area under a curve where the interval is infinite or the integrand has a vertical asymptote. The method involves replacing the problematic bound with a variable limit, evaluating the resulting proper integral, and then taking the limit as the variable approaches the original bound.

The setup

Identify the type of improper integral. Type 1 has infinite intervals, such as limits from $a$ to $\infty$ , $-\infty$ to $b$ , or $-\infty$ to $\infty$ . Type 2 has a discontinuous integrand at the bounds or within the interval. Replace the problematic bound with a variable, typically $t$ or $R$ .

The steps

Substitute the infinite or discontinuous limit with a variable $t$ . 2. Place the limit operator $\lim_{t o c}$ outside the integral. 3. Integrate the function using standard techniques to find the antiderivative. 4. Evaluate the proper integral using the Fundamental Theorem of Calculus. 5. Evaluate the limit. If the limit is a finite number, the integral converges to that value. If the limit is infinite or does not exist, the integral diverges.

Checking the result

Verify the antiderivative by differentiating it back to the original integrand. When taking the limit, check for indeterminate forms and apply L'Hopital's Rule if necessary. Ensure the sign of your result aligns with the behavior of the integrand over the interval (e.g., a positive function must yield a positive converging area).

Common errors

Failing to split the integral when a vertical asymptote occurs strictly between the bounds. Treating $\infty$ algebraically as a number instead of writing out the limit. Incorrectly evaluating limits of logarithmic or exponential functions, such as assuming $\ln(x)$ converges as $x o \infty$ .

Worked example

Evaluate $\int_{1}^{\infty} \frac{1}{x^2} dx$ .

Set up the limit: $\lim_{t o \infty} \int_{1}^{t} x^{-2} dx$ . Find the antiderivative: $\int x^{-2} dx = -x^{-1}$ . Evaluate the definite integral: $\left[ -\frac{1}{x} \right]_{1}^{t} = -\frac{1}{t} - (-\frac{1}{1}) = 1 - \frac{1}{t}$ . Take the limit: $\lim_{t o \infty} (1 - \frac{1}{t}) = 1 - 0 = 1$ . The integral converges to $1$ .

FAQ

Run your own problem

Answer form

References: Stewart Calculus, Section 7.8 · OpenStax Calculus Volume 2, Chapter 3.7