Limited Integration Calculator: Compute Definite Integrals Step‑by‑Step
What it does
- Computes definite integrals (integrals with upper and lower limits) both symbolically and numerically.
- Shows each step of the evaluation: antiderivative, application of the Fundamental Theorem of Calculus, substitution of limits, and final numeric result.
- Handles common techniques automatically: u‑substitution, integration by parts, basic trigonometric and exponential integrals, and simple rational-function integrals (partial fractions).
Input it accepts
- Function of one variable (e.g., sin(x), x^2e^x, 1/(x^2+1))
- Lower and upper limits (numeric or symbolic where supported)
- Option flags: numeric precision, show steps on/off, method hints (prefer substitution/parts), symbolic-only or numeric-only
Output it provides
- Step‑by‑step solution:
- Identify or compute an antiderivative F(x).
- Evaluate F(upper) − F(lower).
- Simplify expression and give numeric value (if requested).
- Error estimates for numeric integration (if using quadrature) and warnings when integral diverges or requires improper integral handling.
- Alternative solution path when multiple methods apply (e.g., substitution vs. parts).
Limitations
- Complex multivariable integrals and advanced special functions may be unsupported.
- Improper integrals need flagged limits; automatic convergence checks are basic.
- Very high-precision symbolic simplification may fail for contrived expressions.
Example
Input: integrate x * e^(x^2) from 0 to 1, show steps. Steps shown:
- Recognize u = x^2 ⇒ du = 2x dx.
- Rewrite integral = ⁄2 ∫ e^u du.
- Antiderivative = ⁄2 e^u.
- Evaluate from 0 to 1 ⇒ ⁄2(e^1 − e^0) = ⁄2(e − 1).
Usage tips
- For numeric-only needs, set a precision and turn off step display for speed.
- Provide hints (e.g., “use substitution”) when integrand is designed for a specific method.
- For improper integrals, supply limit notation (e.g., integrate 1/x^2 from 1 to ∞).
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