Integral Step-by-Step Calculator

Calculator inputs

Supported patterns: 3*x^2 - 4*x + 7, sin(2x), cos(x), exp(3x), ln(x)

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Expression *

Use x (or your variable), *, ^, and parentheses.

Variable

Letters only. Default is x.

Integral type

Choose indefinite or definite integration.

Mode

Numeric mode requires bounds.

Lower bound (a)

Upper bound (b)

Result appears above after submitting.

Plotly graph

The chart shows the integrand f(x) and a visual cumulative area estimate. When available, it also plots the antiderivative F(x).

Range: - to -

The cumulative curve is a visualization aid, not a proof step.

Example data table

#	Expression	Type	Bounds	Mode	Sample result
1	3x^2 - 4x + 7	Indefinite	—	Symbolic	x^3 - 2*x^2 + 7x + C
2	sin(2x) + cos(x)	Indefinite	—	Symbolic	-0.5*cos(2x) + sin(x) + C
3	exp(3x)	Definite	[0, 1]	Symbolic	(e^3 - 1)/3
4	ln(x)	Definite	[1, 2]	Numeric	≈ 0.38629

These rows are examples to help you format inputs.

Recent calculations

No history yet. Run a calculation to populate this list.

Use CSV/PDF buttons after a run to export.

Formula used

Linearity: ∫(f + g) dx = ∫f dx + ∫g dx
Power rule: ∫ axⁿ dx = axⁿ⁺¹/(n+1) + C, for n ≠ −1
Log rule: ∫ a/x dx = a ln|x| + C
Trig rules: ∫ a sin(kx) dx = −a/k · cos(kx) + C, ∫ a cos(kx) dx = a/k · sin(kx) + C
Exponential rule: ∫ a e^kx dx = a/k · e^kx + C
Fundamental theorem: ∫_a^b f(x) dx = F(b) − F(a)
Simpson’s rule (numeric mode): approximate ∫_a^b f(x) dx using weighted samples on evenly spaced points.

This calculator applies rule-based steps to supported patterns, then evaluates bounds when requested.

How to use this calculator

Type your expression using * for multiplication and ^ for powers.
Select the variable symbol, usually x.
Choose Indefinite for an antiderivative, or Definite to compute a numeric value.
Pick Symbolic rules to see step-by-step integration, or Numeric for Simpson’s approximation.
If using definite integration, enter both bounds, then press Submit.
Use Download CSV or Download PDF to export your latest work and history.

Tip: If a term is marked unsupported, rewrite it into polynomials, sin/cos, exp, or ln terms.

Rule coverage and input reliability

The calculator applies linearity first, then integrates each term using standard rules. For polynomial terms, powers are restricted to integer exponents and the special case x^-1. For trig and exponential forms, constants are handled through k-scaling to keep steps readable.

Symbolic workflow for faster checking

In symbolic mode, the output includes each rule application so you can verify intermediate transformations. Typical classroom checks include confirming coefficients, verifying exponent increments, and ensuring divisions by (n+1) and k are applied correctly. This makes it suitable for homework review and quick revision.

Definite integrals and numerical validation

When bounds are supplied, the tool evaluates F(b) − F(a) to produce the definite value. If symbolic evaluation is not possible, numeric mode uses Simpson’s rule with 400 subintervals. That setting balances stability and speed for smooth functions within common study ranges.

Data exports for reporting and practice logs

Each successful run is saved in the session history and can be exported. CSV is useful for building a practice dataset, while the PDF summary captures the most recent expression, settings, and final answer. Keeping a log helps compare results across different bounds and methods.

Graph interpretation for learning

The Plotly chart visualizes f(x) across the chosen range and overlays a cumulative area curve. For definite integrals, the filled region indicates sign and magnitude changes. When an antiderivative is available, plotting F(x) provides an intuitive check on slope behavior.

Recommended input patterns and study tips

Best performance comes from expanding expressions into sums of supported terms such as axⁿ, sin(kx), cos(kx), exp(kx), and ln(x). Break complex formulas into simpler pieces, integrate each, and then recombine. This mirrors exam technique and reduces algebra mistakes.

FAQs

1) Which expressions are supported in symbolic mode?

Symbolic steps handle constants, ax, axⁿ (integer n), x^-1, sin(kx), cos(kx), exp(kx), e^(kx), and ln(x). Write sums of these terms for best results.

2) Why does a term show “Unsupported term”?

The step engine is rule-based and expects standard patterns. If your input uses products, quotients, or nested functions, expand or rewrite it into a sum of supported terms, or switch to numeric mode with bounds.

3) When should I use numeric mode?

Use numeric mode when you only need a definite value and your expression is hard to rewrite. Numeric mode requires bounds and uses Simpson’s rule, which is accurate for many smooth functions over typical intervals.

4) How is the definite integral computed in symbolic mode?

The tool first finds an antiderivative F(x). It then evaluates F(b) and F(a) and subtracts them using the fundamental theorem. The step list shows both endpoint values for quick verification.

5) Does the Plotly graph affect the result?

No. The graph is a learning aid that samples f(x) and a cumulative area estimate for visualization. Your displayed answer comes from symbolic evaluation or the numeric Simpson calculation, not the plot sampling.

6) What should I export in CSV versus PDF?

Export CSV when you want a list of multiple runs for tracking practice or sharing datasets. Export PDF when you need a single summary of the latest calculation for notes, assignments, or quick reporting.