Improper Integrals Formula

Improper integrals are integrals where the interval of integration is infinite (Type I: _a^ f(x)\,dx) or the integrand has an infinite discontinuity on.

The Formula

Type I: ∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx\int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx
Type II: ∫abf(x) dx=lim⁡c→a+∫cbf(x) dx\int_a^b f(x)\,dx = \lim_{c \to a^+} \int_c^b f(x)\,dx (if ff is unbounded at aa)

When to use: Can an infinite region have a finite area? Surprisingly, yes. The area under 1x2\frac{1}{x^2} from 1 to infinity is exactly 1. Improper integrals extend integration to infinite intervals and unbounded functions by using limits to handle the 'improper' part.

Quick Example

∫1∞1x2 dx=lim⁡b→∞∫1b1x2 dx=lim⁡b→∞[−1x]1b=lim⁡b→∞(−1b+1)=1\int_1^{\infty} \frac{1}{x^2}\,dx = \lim_{b \to \infty} \int_1^b \frac{1}{x^2}\,dx = \lim_{b \to \infty} \left[-\frac{1}{x}\right]_1^b = \lim_{b \to \infty} \left(-\frac{1}{b} + 1\right) = 1
Converges to 1.

Notation

Type I: infinite interval (∫a∞\int_a^{\infty}, ∫−∞b\int_{-\infty}^b, ∫−∞∞\int_{-\infty}^{\infty}). Type II: infinite integrand (discontinuity inside [a,b][a,b]).

What This Formula Means

Integrals where the interval of integration is infinite (Type I: ∫a∞f(x) dx\int_a^{\infty} f(x)\,dx) or the integrand has an infinite discontinuity on the interval (Type II: ∫abf(x) dx\int_a^b f(x)\,dx where ff blows up at some point in [a,b][a, b]). Evaluated as limits of proper integrals.

Can an infinite region have a finite area? Surprisingly, yes. The area under 1x2\frac{1}{x^2} from 1 to infinity is exactly 1. Improper integrals extend integration to infinite intervals and unbounded functions by using limits to handle the 'improper' part.

Formal View

Type I: ∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx\int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx. Type II: if lim⁡x→a+f(x)=±∞\lim_{x \to a^+} f(x) = \pm\infty, then ∫abf(x) dx=lim⁡c→a+∫cbf(x) dx\int_a^b f(x)\,dx = \lim_{c \to a^+} \int_c^b f(x)\,dx. The integral converges if the limit is finite; it diverges otherwise.

Worked Examples

Example 1

easy
Evaluate ∫1∞1x2 dx\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx.

Answer

11

First step

1
Replace the infinite upper limit with a variable: ∫1∞1x2 dx=lim⁡b→∞∫1bx−2 dx\int_1^{\infty}\frac{1}{x^2}\,dx = \lim_{b\to\infty}\int_1^b x^{-2}\,dx

Full solution

  1. 2
    Integrate x−2x^{-2}: =lim⁡b→∞[−1x]1b=lim⁡b→∞(−1b+1)= \lim_{b\to\infty}\left[-\frac{1}{x}\right]_1^b = \lim_{b\to\infty}\left(-\frac{1}{b} + 1\right)
  2. 3
    Take the limit as b→∞b \to \infty: since 1b→0\frac{1}{b} \to 0, the integral converges to 11.
Replace ∞\infty with bb, integrate, take the limit. The 1/b1/b term vanishes.

Example 2

hard
Evaluate ∫011x dx\displaystyle\int_0^1 \frac{1}{\sqrt{x}}\,dx (Type II).

Example 3

medium
Evaluate ∫0∞xe−2x dx\int_0^{\infty}xe^{-2x}\,dx.

Common Mistakes

  • Treating ∞\infty as a number to plug in - replace the infinite bound with a variable and take its limit explicitly.
  • Missing an interior asymptote - if the integrand blows up INSIDE [a,b][a,b], split the integral at that point (Type II), don't integrate across it.
  • Reporting an answer for a divergent integral - if the limit is infinite or fails to exist, the integral diverges (no finite value).

Why This Formula Matters

It extends area and accumulation to unbounded settings — probability densities, total work to escape gravity, present value of a perpetuity — where the region is infinite yet the total can be finite. The convergence question (does this infinite area add up?) is the integral analogue of series convergence. Recognizing it by "Does this integral run to infinity or pass through a point where the integrand blows up?" — rather than by familiar numbers — is what lets a student tell it apart from proper definite integral and series convergence and indeterminate-form limit in a mixed problem set.

Frequently Asked Questions

What is the Improper Integrals formula?

Integrals where the interval of integration is infinite (Type I: ∫a∞f(x) dx\int_a^{\infty} f(x)\,dx) or the integrand has an infinite discontinuity on the interval (Type II: ∫abf(x) dx\int_a^b f(x)\,dx where ff blows up at some point in [a,b][a, b]). Evaluated as limits of proper integrals.

How do you use the Improper Integrals formula?

Can an infinite region have a finite area? Surprisingly, yes. The area under 1x2\frac{1}{x^2} from 1 to infinity is exactly 1. Improper integrals extend integration to infinite intervals and unbounded functions by using limits to handle the 'improper' part.

What do the symbols mean in the Improper Integrals formula?

Type I: infinite interval (∫a∞\int_a^{\infty}, ∫−∞b\int_{-\infty}^b, ∫−∞∞\int_{-\infty}^{\infty}). Type II: infinite integrand (discontinuity inside [a,b][a,b]).

Why is the Improper Integrals formula important in Math?

It extends area and accumulation to unbounded settings — probability densities, total work to escape gravity, present value of a perpetuity — where the region is infinite yet the total can be finite. The convergence question (does this infinite area add up?) is the integral analogue of series convergence. Recognizing it by "Does this integral run to infinity or pass through a point where the integrand blows up?" — rather than by familiar numbers — is what lets a student tell it apart from proper definite integral and series convergence and indeterminate-form limit in a mixed problem set.

What do students get wrong about Improper Integrals?

The procedure for improper integrals is the easy part; the trap is treating ∞\infty as a number to plug in. Asking "Does this integral run to infinity or pass through a point where the integrand blows up?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Improper Integrals formula?

Before studying the Improper Integrals formula, you should understand: definite integral, limit, infinity.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Partial Fraction Decomposition: Step-by-Step Guide →
← Back to Improper Integrals See All Examples Practice Problems