Improper Integral Calculator

Evaluate integrals with infinite limits or unbounded integrands. Step‑by‑step symbolic solution and convergence analysis.

Improper integral types: Type I (infinite limits)   |   Type II (discontinuities)

The calculator attempts symbolic integration and evaluates limits at singularities at the endpoints only. For interior discontinuities, split the integral manually.

Use standard math notation: ^ for exponent, * for multiplication, functions: sin, cos, exp, log, sqrt, etc.
1/x² from 1 to ∞
1/x from 1 to ∞
e⁻ˣ from 0 to ∞
1/√x from 0 to 1
1/(x²+1) from -∞ to ∞
ln(x) from 0 to 1
1/(x-1)² from 1 to 2
Use -inf for −∞
Use inf for +∞
Computing symbolic integral & limits...

Understanding Improper Integrals

An improper integral occurs when the interval of integration is infinite or the integrand has an infinite discontinuity within the interval. Convergence means the area under the curve approaches a finite number.

Type I – Infinite limits

\[ \int_a^\infty f(x)\,dx = \lim_{t\to\infty}\int_a^t f(x)\,dx \]

\[ \int_{-\infty}^b f(x)\,dx = \lim_{t\to-\infty}\int_t^b f(x)\,dx \]

\[ \int_{-\infty}^\infty f(x)\,dx = \int_{-\infty}^c f(x)\,dx + \int_c^\infty f(x)\,dx \] (both parts must converge)

Type II – Discontinuities at endpoints or interior

If f has a vertical asymptote at x = c (a ≤ c ≤ b):

\[ \int_a^b f(x)\,dx = \lim_{t\to c^-}\int_a^t f(x)\,dx + \lim_{t\to c^+}\int_t^b f(x)\,dx \]

Note: This calculator only checks for discontinuities at the bounds a and b. For interior singularities (e.g., ∫₋₁¹ 1/x² dx), you must split the integral manually.

Convergence Tests (p‑test)

Integral form Convergence condition Divergence condition
∫₁^∞ 1/xp dx p > 1 p ≤ 1
∫₀¹ 1/xp dx p < 1 p ≥ 1
∫₀^∞ e-ax dx (a>0) always converges to 1/a –
∫₂^∞ 1/(x (ln x)p) dx p > 1 p ≤ 1

Step‑by‑Step Approach

1

Identify the type: Are limits infinite? Does the integrand blow up at an endpoint or inside?

2

Find antiderivative F(x) using symbolic integration.

3

Replace improper limits with limits: e.g., limt→∞ F(t) – F(a) for infinite upper bound.

4

Evaluate the limits. If both one‑sided limits exist and are finite, the integral converges; otherwise it diverges.

Calculator features (updated):

  • Symbolic integration with nerdamer (supports polynomials, exponentials, trig, logs)
  • Automatic limit detection for inf, -inf and endpoint singularities
  • Correct handling of infinite limit results (now returns ±∞ instead of error)
  • Handles symbolic constants (like π) in limit results and converts them to numbers
  • Graph of integrand for visual reference
  • Convergence/divergence indication with exact value when finite
Important limitation: This tool does not automatically detect or split integrals with discontinuities inside the integration interval (e.g., ∫₋₁¹ 1/x² dx). For such cases, you must manually split the integral at the singular point and evaluate each part separately.

Frequently Asked Questions

An integral is improper if either the interval is unbounded (e.g., [a, ∞)) or the integrand becomes infinite within the interval (e.g., 1/x near 0). These require limits to define.

Type inf or ∞ for +∞, and -inf or -∞ for −∞. The calculator will treat them as limits.

Currently, the calculator only checks for singularities at the bounds. If the integrand has a vertical asymptote inside (e.g., at x=0 for ∫₋₁¹ 1/x² dx), the result will be incorrect. You must split the integral manually at the singular point.

The calculator will show a warning. In such cases, you may need to use numerical methods or comparison tests (not implemented here).

For convergent improper integrals, the result is given exactly (symbolically) if possible. No numerical approximation is performed; focus is on exact analysis.